Duality for second-order symmetric multiobjective ... - Semantic Scholar
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Int. J. Mathematics in Operational Research, Vol. 4, No. 2, 2012
Duality for second-order symmetric multiobjective programming with cone constraints S.K. Gupta* and D. Dangar Department of Mathematics, Indian Institute of Technology Patna, Patna 800 013, Bihar, India Fax: +91-612-2277384 E-mail: [email protected] E-mail: [email protected] *Corresponding author Abstract: In this paper, a new pair of Mond-Weir type multiobjective second-order symmetric dual models with cone constraints is formulated in which the objective function is optimised with respect to an arbitrary closed convex cone. Usual duality relations are further established under K-η-bonvexity/second-order symmetric dual K-H-convexity assumptions. A nontrivial example has also been illustrated to justify the weak duality theorems. Several results including many recent works are obtained as special cases. Keywords: symmetric duality; K-η-bonvexity; second-order K-H-convexity; multiobjective programming; cones. Reference to this paper should be made as follows: Gupta, S.K. and Dangar, D. (2012) ‘Duality for second-order symmetric multiobjective programming with cone constraints’, Int. J. Mathematics in Operational Research, Vol. 4, No. 2, pp.128–151. Biographical notes: S.K. Gupta is an Assistant Professor in the Department of Mathematics, Indian Institute of Technology Patna, India. He obtained his MSc and PhD from Indian Institute of Technology Roorkee, India. He has published many papers in the area of Mathematical Programming. His research interests include fuzzy optimisation and support vector machines. D. Dangar recently is a Research Scholar in the Department of Mathematics, Indian Institute of Technology Patna, India. He obtained his Bachelors Degree from University of Burdwan, West Bengal, India in 2007 and Masters Degree from Banaras Hindu University, Varanasi, India in 2009.
1
Introduction
Mangasarian (1975) introduced the concept of second-order duality in nonlinear programming. Mond (1974) gave an idea of second-order convex function, which was named as bonvex function by Bector and Chandra (1985). Pandey (1991) introduced the concept of η-bonvex functions as a generalisation of bonvex functions. Yang and Hou (2001) studied a pair of second-order symmetric nondifferentiable dual programs and Copyright © 2012 Inderscience Enterprises Ltd.
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proved the duality results under F-pseudoconvexity assumptions. Alidaee et al. (2009), studied some nonlinear models and their computational significance for combinatorial optimisation problems. Wolfe and Mond-Weir type multiobjective symmetric dual programs has been discussed in Kim and Kim (2008) and proved appropriate duality relations under cone-invexity/cone-pseudoinvexity assumptions. Gulati et al. (2010) presented a pair of Wolfe and Mond-Weir type second-order symmetric multiobjective dual problems over arbitrary cones and proved weak, strong and converse duality theorems under η-bonvexity/η-pseudobonvexity assumptions. Ahmad and Husain (2010a) proved duality results under second-order invexity assumptions for a pair of Wolfe type second-order multiobjective symmetric dual programs with arbitrary cone constraints. Saini and Gulati (2011) formulated a pair of Wolfe type multiobjective second-order symmetric dual programs over arbitrary cones for nondifferentiable functions and proved duality theorems under second-order K-F-convexity assumptions. Usual duality relations has been proved in Gupta and Kailey (2011) for a pair of Wolfe type nondifferentiable second-order multiobjective dual programs under F-convexity assumptions. Gulati et al. (2011) discussed the duality results for Wolfe and Mond-Weir type nondifferentiable multiobjective symmetric dual programs over arbitrary cones under K-preinvexity/ K-convexity/pseudoinvexity assumptions. The work in Agarwal et al. (2011) fills some gap in the work of Chen (2004) by giving the correct proof of strong duality theorem for higher-order Mond-Weir type multiobjective nondifferentiable symmetric dual programs. In this paper, we consider a new pair of Mond-Weir type second-order multiobjective symmetric dual programs over arbitrary cones and proved appropriate duality relations under K-η-bonvexity/second-order K-H-convexity assumptions. We also illustrate a nontrivial example to verify our weak duality theorems.
2
Literature review
In mathematical programming, a primal-dual pair is said to be symmetric if the dual of the dual is a primal problem that is when the dual can be recast in the form of primal. Unlike linear programming, the majority of dual formulation in nonlinear programming do not possess the symmetric property. For instance, the duals formulated by Wolfe (1961) and Mond and Weir (1999) are not symmetric. Dorn (1960) first introduced the concept of symmetric duality in quadratic programming. Introducing a differentiable function ψ: Rn × Rm → R, the work in Dorn (1960) was later on extended to following nonlinear symmetric dual programs by Dantzig et al. (1965): (P1)
Minimise ψ ( x, y ) − yT ∇ yψ ( x, y ) subject to ∇ yψ ( x, y ) ≤ 0, x ≥ 0, y ≥ 0.
(D1)
Maximise ψ (u , v) − u T ∇ xψ (u, v) subject to ∇ xψ (u , v) ≥ 0, u ≥ 0, v ≥ 0.
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To weaken convexity-concavity assumptions on ψ(x, y) to pseudoconvexitypseudoconcavity, Weir and Mond (1988) studied the following pair of symmetric dual programs: (P2)
Minimise ψ ( x, y ) subject to ∇ yψ ( x, y ) ≤ 0, yT ∇ yψ ( x, y ) ≥ 0 x ≥ 0.
(D2)
Maximise ψ (u , v) subject to ∇ xψ (u , v) ≤ 0, u T ∇ xψ (u , v) ≤ 0 v ≥ 0.
Bazaraa and Goode (1973) established symmetric duality results for functions with arbitrary cones. Such a formulations enables one to consider infinitely many constraints of the inequality type. Motivated by Bazaraa and Goode (1973) and Weir and Mond (1988), Chandra and Kumar (1998) formulated the following Mond-Weir type symmetric dual programs over arbitrary cones and proved the results under pseudoinvexity type assumptions: (P3)
Minimise ψ ( x, y ) subject to ∇ yψ ( x, y ) ∈ C2* , y T ∇ yψ ( x, y ) ≥ 0, x ∈ C1 .
(D3)
Maximise ψ (u, v) subject to − ∇ xψ (u, v) ∈ C1* , u T ∇ xψ (u, v) ≤ 0, v ∈ C2 ,
where C1* and C2* are the polar cones of closed convex cones C1 and C2, respectively. Hanson and Mond (1982) formulated a type of generalised convexity and established duality relations between the general nonlinear programming problem and its Wolfe dual. Gulati and Craven (1983) and Mond and Egudo (1985) gave a strict converse duality in nonlinear programming. Egudo and Mond (1986) established duality theorems between the nonlinear programming problem and its Mond-Weir dual under F-quasiconvex/pseudoconvex assumptions and also generalise the strict converse duality theorem given in Gulati and Craven (1983) and Mond and Egudo (1985). A second-order dual for a nonlinear program with linear constraints in complex space over arbitrary polyhedral convex cones is presented and usual duality results are established in Gupta (1983). Suneja and Gupta (1998) described Wolfe and Mond-Weir type duals for a multiobjective nonlinear programming problem involving semilocally convex and related functions in terms of their right differentials and proved duality results under semilocal convexity/quasiconvexity/pseudoconvexity assumptions.
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Gulati et al. (1997) have formulated Wolfe and Mond-Weir type multiobjective symmetric dual problems and proved usual duality results under invexity/generalised invexity assumptions without the nonnegativity restrictions. Chandra and Kumar (1998) has taken a Mond-Weir type symmetric dual programs over arbitrary cones and established duality relations under pseudoinvexity assumptions. Chan and Sun (2008) reported for semidefinite programming that the primal/dual constraint non-degeneracy is equivalent to the dual/primal strong second-order sufficient condition. Following the work in Chan and Sun (2008), Qi (2009) discussed duality results in nonlinear semidefinite programming. Nonlinear optimisation problems have many applications like quadratic programming problems are used in portfolio optimisation, inventory management, engineering design etc. The performance of a given portfolio policy can be evaluated by comparing its expected utility with that of the optimal policy, which in general is not computable, in that case a direct comparison is not possible. Haugh et al. (2006) solved this problem by using the given portfolio policy to the construction based on a dual formulation of the portfolio optimisation problem. Multiobjective problems have vast number of applications in practical real life situations like multicriteria decision making approach, which is used for sensitivity analysis in analytic hierarchy process (see Sowlati et al., 2010), portfolio optimisation (see Xidonas et al., 2010), goal programming, risk programming etc. A pair of second-order Lagrangian primal-dual algorithm for inequality constrained optimisation problems that generates a sequence converging to points satisfying the second-order necessary conditions for optimality has discussed by Pillo et al. (2005). Mordukhovich (2007) devoted his study to apply the modern methods of variational analysis to constrained optimisation and control problems. Actually his main focus was on the discussion of problems with nonsmooth structures. Zheng and Yang (2007) provided Lagrange multiplier rules for a class of semi-infinite optimisation problems where all functions are lower semicontinuous using the variational analysis technique. Schachinger and Bomze (2009) described optimisation of quadratic functions over a polyhedron and applied the results to establish a Frank-Wolfe-type theorem for the primal-dual pair of a class of conic programs which shows the existence of the solution of the dual. Suneja et al. (2003) proved weak, strong and converse duality theorems under η-bonvexity/η-pseudobonvexity assumptions for the Mond-Weir type second-order symmetric multiobjective dual models without nonnegativity constraints. Ahmad and Husain (2005) and Ahmad (2005a) considered a slightly different type of Mond-Weir nondifferentiable multiobjective second-order symmetric dual programs and proved appropriate duality results under F-pseudoconvexity/F-pseudoconcavity and η-bonvexity/η-boncavity, assumptions, respectively. Gulati and Gupta (2005) studied a pair of Wolfe type second-order symmetric dual programs with nondifferentiable functions and proved duality theorems under bonvexity/boncavity assumptions. Second-order symmetric duality for nondifferentiable multiobjective programs has been taken in Ahmad (2005a) and proved duality theorems under η-pseudobonvexity assumptions. Gulati et al. (2008) proved weak, strong and converse duality theorems for Wolfe and Mond-Weir type second-order symmetric duals with differentiable functions over arbitrary cones under the η-bonvexity/ η-pseudobonvexity assumptions. A pair of Mond-Weir type multiobjective second-order symmetric duality over cones have described in Gulati and Mehndiratta (2010b) for differentiable functions proved duality relations under K-F-convexity/K-η-bonvexity
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assumptions. Motivated by Bector et al. (1999), Ahmad (2005b), Suneja et al. (2002) and Khurana (2005), Ahmad and Husain (2010b) formulate a pair of multiobjective mixed symmetric dual programs over arbitrary cones and derived duality results under cone-invexity/pseudoinvexity assumptions. Mond (1978) studied nonlinear fractional programming problems and further established necessary and sufficient conditions for optimality. Ahmad et al. (2011) presented a second-order dual for a nondifferentiable fractional programming problem which consists of maximising the ratio of functions involving square root terms of positive semidefinite quadratic forms and established duality results using second-order (F, α, ρ, d)-convexity assumptions. Gupta and Jayswal (2010) gave a Mond-Weir type higher-order multiobjective symmetric dual programs over arbitrary cones and proved duality results under higher-order cone-invexity/pseudoinvexity assumptions. Preda et al. (2011) established duality results for a fractional programming problem by replacing convexity/sublinearity assumptions on F by quasiconvexity. A class of nondifferentiable minimax programming problems has been discussed and further duality results are established in Jayswal and Stancu-Minasian (2011). Kim and Lee (2009) considered Wolfe and Mond-Weir type higher-order multiobjective dual programs over arbitrary cones and established duality relations under higher-order pseudo-type I/(F, ρ)-type I assumptions.
3
Notations and definitions
Consider the following multiobjective programming problem: (P )
K -Minimise ξ (x) subject to x ∈ X ° = {x ∈ S : −φ ( x) ∈ Q},
where S ⊆ Rn be open, ξ: S → Rk, φ: S → Rm, K is closed convex pointed cone and Q is closed convex cone with nonempty interiors in Rk and Rm, respectively. Definition 1 (Gulati and Mehndiratta, 2010b): A point x ∈ X° is an efficient solution of (P) if there exists no x ∈ X° such that
ξ ( x ) − ξ ( x) ∈ K \ {0}. Definition 2 (Chandra and Kumar, 1998): A set C of Rn is called a cone, if for each x ∈ C and λ ∈ R, λ 0 0 , we have λx ∈ C. Moreover, if C is convex then it is called convex cone. Definition 3 (Suneja et al., 2002): The positive polar cone C* of a cone C is defined by C * = {z ∈ R n : xT z ≥ 0 for all x ∈ C}.
Let T be a closed convex cone in Rn, with non-empty interior. Let S ⊆ Rn be an open set such that T ⊂ S.
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Definition 4 (Gulati and Mehndiratta, 2010a): A twice differentiable function ξ : S → Rk is said to be K-η-bonvex at u ∈ S if there exists η : S × S → Rn such that for all (x, ri) ∈ S × Rn, i = 1, 2, …, k, ξ1 ( x) − ξ1 (u ) + 1/ 2r1T ∇ xxξ1 (u )r1 − η T ( x, u )(∇ xξ1 (u ) + ∇ xxξ1 (u )r1 ),… , ∈ K . T T ξ k ( x) − ξ k (u ) + 1/ 2rk ∇ xxξ k (u )rk − η ( x, u )(∇ xξ k (u ) + ∇ xxξ k (u )rk )
Definition 5 (Ahmad and Husain, 2005): A functional H: S × S × Rn → R is said to be sublinear with respect to the third variable if for all (x, u) ∈ S × S, •
H(x, u; a1 + a2) / H(x, u; a1) + H(x, u; a2) for all a1, a2 ∈ Rn
•
H(x, u; αa) = αH(x, u; a), for all α ∈ R+ and for all a ∈ Rn.
For notational convenience, we write H(x, u; a) = Hx,u(a). Definition 6 (Gulati and Mehndiratta, 2010a): Let H be a sublinear functional with respect to the third variable and ξ : S → Rk be a twice differentiable function. Then ξ is said to be second-order K-H-convex at u ∈ S, if for all (x, ri) ∈ S × Rn, i = 1, 2, ..., k, ξ1 ( x) − ξ1 (u ) + 1/ 2r1T ∇ xxξ1 (u )r1 − H x ,u (∇ xξ1 (u ) + ∇ xxξ1 (u )r1 ),… , ∈ K. ξ ( x) − ξ (u ) + 1/ 2r T ∇ ξ (u )r − H (∇ ξ (u ) + ∇ ξ (u )r ) k k xx k k x ,u x k xx k k k
Throughout this paper, we have taken p = (p1, …, pk), q = (q1, …, ql), r = (r1, …, rk), s = (s1, …, sl), λ = (λ1, …, λk) and w = (w1, …, wl), where pi, qj ∈ Rm, ri, sj ∈ Rn, λi, wj ∈ R for i = 1, 2, …, k, j = 1, 2, …, l.
4
Second-order symmetric duality
Suneja et al. (2002), formulated a pair of multiobjective symmetric dual programs over arbitrary cones and established weak, strong, converse and self duality theorems under K-convexity/K-concavity assumptions. Khurana (2005) proved duality theorems under K-pseudoinvexity assumptions for a pair of Mond-Weir type dual programs. Yang et al. (2005) established duality relations under F-convexity assumptions for the Mond-Weir type second-order symmetric nondifferentiable multiobjective dual problems. Kassem (2006) presented a pair of symmetric multiobjective dual programs and proved duality relations under convexity/concavity assumptions. Gulati and Mehndiratta (2009) pointed out certain omissions in Kaseem (2006) and Kim et al. (1997) and gave corrected models and proofs of their duality theorems. Recently Gulati et al. (2010) formulated Wolfe and Mond-Weir type second-order multiobjective symmetric dual for differentiable functions and Gulati and Mehndiratta (2010a) considered for nondifferentiable case. Further, they proved duality theorems under η-bonvexity/η-pseudobonvexity and K-η-bonvexity/ K-F-convexity assumptions, respectively. Inspired by above mentioned papers, we formulate the following pair of Mond-Weir type second-order multiobjective symmetric dual programs over arbitrary cones:
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Primal (MSP) K - mi nimise L( x, y, λ , w, p, q ) = { f1 ( x, y ) − 1/ 2 p1T ∇ yy f1 ( x, y ) p1 ,… , f k ( x, y ) − 1/ 2 pkT ∇ yy f k ( x, y ) pk }
subject to l k − ∑ λi ( ∇ y fi ( x, y ) + ∇ yy fi ( x, y ) pi ) + ∑ w j ∇ yy g j ( x, y )q j ∈ C2* , j =1 i =1
(1)
l k yT ∑ λi ( ∇ y f i ( x, y ) + ∇ yy f i ( x, y ) pi ) + ∑ w j ∇ yy g j ( x, y )q j ≥ 0, j =1 i =1
(2)
λ ∈ int K * , x ∈ C1 , w j ≥ 0, j = 1, 2, … , l ,
(3)
Dual (MSD) K -maximise M (u , v, λ , w, r , s ) = { f1 (u, v) − 1/ 2r1T ∇ xx f1 (u , v)r1 ,… , f k (u , v) − 1/ 2rkT ∇ xx f k (u , v)rk }
subject to k
∑ λ (∇ i
i =1
l
x
fi (u, v) + ∇ xx fi (u , v)ri ) + ∑ w j ∇ xx g j (u, v) s j ∈ C1* ,
(4)
j =1
l k u T ∑ λi ( ∇ x f i (u , v) + ∇ xx f i (u, v)ri ) + ∑ w j ∇ xx g j (u , v) s j ≤ 0, j =1 i =1
(5)
λ ∈ int K * , v ∈ C2 , w j ≥ 0, j = 1, 2, … , l ,
(6)
where for i = 1, 2, …, k, j = 1, 2, …,l, •
C1 and C2 are closed convex cone with non-empty interior in Rn and Rm, respectively,
•
S1 ⊆ Rn and S2 ⊆ Rm are open sets such that C1 × C2 ⊂ S1 × S2,
•
fi : S1 × S2 → R and gj : S1 × S2 → R, are twice differentiable functions of x and y
•
C1* , C2* and K * are positive polar cones of Cl, C2 and K, respectively.
Now, we prove the weak, strong, converse duality theorems for dual pair (MSP) and (MSD). Theorem 1 (Weak duality): Let (x, y, λ, w, p, q) and (u, v, λ, w, r, s) be feasible for (MSP) and (MSD), respectively. Suppose there exist functions η1 : Sl × Sl → Rn and η2 : S2 × S2 → Rm satisfying
η1 ( x, u ) + u ∈ C1 ,
(A)
η 2 ( v, y ) + y ∈ C2 ,
(B)
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0 ∇ g (u , v) sj 0 (η1T ( x, u ) η 2T (v, y )) xx j ≤ 0, −∇ yy g j ( x, y ) 0 q j 0 for j = 1,2, … , l.
(C)
Furthermore, let {f1(., v), …, fk (., v)} be K-η1-bonvex at u and –{f1(x,.), …, fk(x,.)} be K-η2-bonvex at y. Then L( x, y, λ , w, p, q) − M (u , v, λ , w, r , s ) ∉ − K \ {0}.
(7)
Proof: By the K-η1-bonvexity of {f1(., v), …, fk(., v)} at u, we have ( f1 ( x, v) − f1 (u , v) + 1/ 2r1T ∇ xx f1 (u , v )r1 − η1T ( x, u )(∇ x f1 (u , v ) + ∇ xx f1 (u , v)r1 ),… , f k ( x, v) − f k (u, v) + 1/ 2rkT ∇ xx f k (u , v )rk − η1T ( x, u )(∇ x f k (u, v) + ∇ xx f k (u , v) rk ) ∈ K .
As λ ∈ int K*, therefore from above we obtain k
∑ λ ( f ( x, v) − f (u, v) + 1/ 2r i
i =1
i
i
T
i
∇ xx f i (u, v)ri
(8)
−η1T ( x, u )(∇ x fi (u , v) + ∇ xx f i (u, v)ri ) ) ≥ 0.
Now, it follows from (A) and dual constraint (4) that
k
l
j =1
(η1 ( x, u ) + u ) ∑ λi ( ∇ x fi (u, v) + ∇ xx fi (u, v)ri ) + ∑ w j ∇ xx g j (u, v) s j ≥ 0, T
i =1
which using inequality (5) implies
k
l
j =1
η1T ( x, u ) ∑ λi ( ∇ x f i (u , v) + ∇ xx f i (u, v)ri ) + ∑ w j ∇ xx g j (u , v) s j ≥ 0. i =1
This further by hypothesis (C) and w j ≥ 0, j = 1, 2, … , l , yields
k
i =1
η1T ( x, u ) ∑ λi ( ∇ x f i (u , v) + ∇ xx fi (u , v)ri ) ≥ 0.
(9)
From inequalities (8) and (9), we get k
∑ λ ( f ( x, v) − f (u, v) + 1/ 2r i
i
i
i =1
i
T
∇ xx f i (u, v)ri ) ≥ 0.
(10)
Similarly, using K-η2-bonvexity of –{f1(x,.), …, fk(x,.)} at y, hypotheses (B), (C) and primal constraints (1) and (2), we obtain k
∑ λ ( f ( x, y) − f ( x, v) − 1/ 2 p ∇ i
i =1
i
i
T i
yy
fi ( x, y ) pi ) ≥ 0.
(11)
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Adding inequalities (10) and (11), we have k
∑ λ ( f ( x, y) − 1/ 2 p i
T i
i
i =1
∇ yy f i ( x, y ) pi − fi (u , v) + 1/ 2riT ∇ xx f i (u, v) ri ) ≥ 0.
(12)
But suppose that L( x, y, λ , w, p, q ) − M (u, v, λ , w, r , s ) ∈ − K \ {0}. i.e.
{( f ( x, y) − 1/ 2 p ∇ T 1
1
f ( x, y ) p1 , …, f k ( x, y ) − 1/ 2 pkT ∇ yy f k ( x, y ) pk )
yy 1
}
−( f1 (u, v) − 1/ 2r1T ∇ xx f1 (u , v)r1 , … , f k (u , v ) − 1/ 2rkT ∇ xx f k (u, v)rk ) ∈ − K \ {0}.
Since λ ∈ int K*, it yields k
∑ λ ( ( f ( x, y) − 1/ 2 p i
i
i =1
T i
∇ yy f i ( x, y ) pi ) − ( fi (u, v) − 1/ 2riT ∇ xx f i (u, v) ri ) ) < 0,
which contradicts inequality (12). Hence the result.
□
Remark 1: If we replace the hypothesis (C) in the above theorem by 0 l ∑ w j ∇ xx g j (u, v) s j l T T (η1 ( x, u ) η 2 (v, y )) j =1 ≤ 0, − ∇ w g x y q ( , ) ∑ j yy j j 0 j =1
then the same conclusion of Theorem 1 also holds. However, taking this hypothesis, the theorem can be obtained without even considering the non-negativity restriction on wj, j = 1, 2, …, l. Theorem 2 (Weak duality): Let (x, y, λ, w, p, q) and (u, v, λ, w, r, s) be feasible for (MSP) and (MSD), respectively. Let for sublinear functionals F: S1 × S1 × Rn → R and G: S2 × S2 × Rm → R satisfying Fx ,u (a ) + u T a ≥ 0
for all a ∈ C1* ,
(A′)
Gv , y (b) + y T b ≥ 0
for all b ∈ C2* ,
(B′)
Fx ,u (∇ xx g j (u , v ) s j ) ≤ 0 and Gv , y (−∇ yy g j ( x, y )q j ) ≤ 0,
for j = 1, 2, …, l.
(C′)
Furthermore, let {f1(., v), …, fk(., v)} be second-order K-F-convex at u, –{f1(x,.), …, fk(x,.)} be second-order K-G-convex at y and R+k ⊆ K . Then L( x, y, λ , w, p, q ) − M (u, v, λ , w, r , s ) ∉ − K \ {0}.
Proof: By second-order K-F-convexity of {f1(., v), …, fk(., v)} at u, we have ( f1 ( x, v) − f1 (u , v) + 1/ 2r1T ∇ xx f1 (u , v)r1 − Fx ,u (∇ x f1 (u, v) + ∇ xx f1 (u, v) r1 ),… , f k ( x, v) − f k (u, v) + 1/ 2rkT ∇ xx f k (u , v)rk − Fx ,u (∇ x f k (u , v) + ∇ xx f k (u, v)rk )) ∈ K .
(13)
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This together with λ ∈ int K* implies k
∑ λ ( f ( x, v) − f (u, v) + 1/ 2r i
i
i
T
i
i =1
∇ xx f i (u, v)ri )
k
≥ ∑ λi Fx ,u ( ∇ x fi (u , v) + ∇ xx f i (u , v)ri ).
(14)
i =1
As R+k ⊆ K ⇒ K * ⊆ R+k and since λ ∈ int K*, therefore
λ > 0.
(15)
Using inequality (15) and sublinearity of F in inequality (14), we have k
∑ λ ( f ( x, v) − f (u, v) + 1/ 2r i
i
i
T
i
i =1
∇ xx f i (u , v)ri )
k ≥ Fx ,u ∑ λi ( ∇ x fi (u, v) + ∇ xx fi (u , v)ri ) . i =1
(16)
Further, it follows from hypothesis (A′) and inequality (4) that l k Fx ,u ∑ λi ( ∇ x fi (u, v) + ∇ xx f i (u , v)ri ) + ∑ w j ∇ xx g j (u, v) s j j =1 i =1 l k +u T ∑ λi ( ∇ x fi (u , v) + ∇ xx f i (u, v)ri ) + ∑ w j ∇ xx g j (u , v ) s j ≥ 0, j =1 i =1 k
l
i =1
j =1
for a = ∑ λi ( ∇ x f i (u, v) + ∇ xx f i (u , v)ri ) + ∑ w j ∇ xx g j (u, v) s j ∈ C1*
which by using inequality (5) implies l k Fx ,u ∑ λi ( ∇ x fi (u , v) + ∇ xx f i (u, v)ri ) + ∑ w j ∇ xx g j (u , v) s j ≥ 0. j =1 i =1
Now, applying sublinearity of F and using w j ≥ 0, for all j, we have k l Fx ,u ∑ λi (∇ x f i (u, v) + ∇ xx fi (u, v)ri ) + ∑ w j Fx ,u (∇ xx g j (u , v) s j ) ≥ 0. i =1 j =1
This using inequality (C′) and w j ≥ 0, j = 1, 2, …, l , we obtain k Fx ,u ∑ λi (∇ x f i (u, v) + ∇ xx fi (u, v)ri ) ≥ 0. i =1
It follows from inequality (16) that k
∑ λ ( f ( x, v) − f (u, v) + 1/ 2r i
i =1
i
i
i
T
∇ xx fi (u , v)ri ) ≥ 0.
(17)
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Similarly, using K-G-convexity of –{f1(x,.), …, fk(x,.)} at y, hypotheses (B′), (C′), (15), primal constraints (1) and (2), and the sublinearity of G, we obtain k
∑ λ ( f ( x, y) − f ( x, y) − 1/ 2 p i
i
T i
i
i =1
∇ yy fi ( x, y ) pi ) ≥ 0.
(18)
Finally adding inequalities (17) and (18), we have k
∑ λ ( f ( x, y) − 1/ 2 p i
T i
i
i =1
∇ yy fi ( x, y ) pi − fi (u , v) + 1/ 2riT ∇ xx f i (u , v)ri ) ≥ 0.
(19)
Now, suppose on the contrary L( x, y, λ , w, p, q ) − M (u, v, λ , w, r , s ) ∈ − K \ {0}. or
{( f ( x, y) − 1/ 2 p ∇ − { f (u , v) − 1/ 2r ∇ 1
1
T 1
yy 1
f ( x, y ) p1 , … , f k ( x, y ) − 1/ 2 pkT ∇ yy f k ( x, y ) pk
T 1
xx 1
}
}
f (u, v)r1 , … , f k (u, v) − 1/ 2rkT ∇ xx f k (u, v)rk ) ∈ − K \ {0}.
Since λ ∈ int K*, therefore we have k
∑ λ ( ( f ( x, y) − 1/ 2 p i
i =1
i
T i
∇ yy f i ( x, y ) pi ) − ( f i (u , v) − 1/ 2riT ∇ xx fi (u , v)ri ) ) < 0,
which contradicts inequality (19). Hence the result.
□
Remark 2: (i) It may be noted that to prove Theorem 2, we also need an additional assumption R+k ⊆ K . (ii) If we replace the hypothesis (C′) in the above theorem by l l Fx ,u ∑ w j ∇ xx g j (u , v) s j ≤ 0 and Gv , y −∑ w j ∇ yy g j ( x, y )q j ≤ 0. j =1 j =1
then the same conclusion of Theorem 2 also holds. Using this assumption in the proof, the requirement of non-negativity restriction on w may be removed. The notations (MSP)λ and (MSD)λ are used in the following theorems to denote (MSP) and (MSD), respectively when λ is fixed to be λ . Theorem 3 (Strong duality): Let ( x , y , λ , w, p, q ) be an efficient solution for ( MSD)λ . Let (i) for i = 1, 2, …, k, ∇ yy f i ( x , y ) be positive definite and
k
∑λ p i
T i
∇ y fi ( x , y ) ≥ 0
j =1
or ∇ yy f i ( x , y ) be negative definite and
k
∑λ p i
T i
∇ y fi ( x , y ) ≤ 0,
j =1
(ii)
the set of vectors {∇yfi ( x , y ) + ∇yyfi ( x , y ) pi }ik=1 be linearly independent,
Duality for second-order symmetric multiobjective programming (iii)
l
∑w ∇ j
yy
j =1
(iv)
139
g j ( x , y )q j ∉ span{∇ y fi ( x , y ) + ∇ yy fi ( x , y ) pi }ik=1 \{0} and
R+k ⊆ K .
Then, pi = 0 for i = 1, 2, ..., k, ( x , y , w, r = 0, s = 0) is feasible for (MSD)λ and the objective function values of (MSP) and (MSD)λ are equal. Furthermore, if the hypotheses of Theorem 1 or Theorem 2 are satisfied for all feasible solutions of (MSP) and (MSD)λ , then ( x , y , w, r , s ) is an efficient solution for (MSD)λ . Proof: Since ( x , y , λ , w, p, q ) is an efficient solution to (MSP), by Fritz John necessary optimality conditions Suneja et al. (2002), there exist α ∈ K*, β ∈ C2, v ∈ R+, δ ∈ R+l , such that the following conditions are satisfied at ( x , y , λ , w, p, q ) (for simplicity, we write ∇xfi, ∇yxfi, ∇yygj instead of ∇ x fi ( x , y ), ∇ yx fi ( x , y ), ∇ yy g j ( x , y ), etc:
(20)
(21) (22) (23) (24) (25)
(26)
(27) (28) (29)
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Inequalities (21) and (22) are equivalent to
(30) and (31) Since ∇yyfi is positive or negative definite, for i = 1, 2, …, k, equation (25) yields (32) From equations (30) and (32), we have
(33) Also, R ⊆ K ⇒ K * ⊆ R and since λ ∈ int K *, therefore k +
k +
λ > 0.
(34)
Now, we claim that αi ≠ 0 for all i = 1, 2, …, k. Otherwise if, αi = 0 for some i, say t0, 1 ≤ t0 ≤ k, then for i = t0, using inequality (34) in equation (32), we get
β = ν y.
(35)
From equations (33) and (35), we have
It follows from hypothesis (iii) that
This using hypothesis (ii), yields
αi = νλi , i = 1, 2, …, k.
(36)
Since λ > 0 and α t0 = 0, therefore from equation (36), ν = 0. Hence equations (35) and (36) implies β = 0 and αi = 0 for all i. Also, from equation (23), δj = 0, j = 1, 2, …, l. Therefore (α, β, v, δ) = 0, which contradicts (29). Hence αi ≠ 0 for all i. Moreover, α ∈ K * ⊆ R+k (by hypothesis (iv)), therefore for i = 1, 2, …, k
αi > 0.
(37)
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Further, using expressions (32) and (37) in equation (31), we have
(38) We now prove that pi = 0, i = 1, 2, …, k. Otherwise, hypothesis (i) implies
which contradicts equation (38) and hence pi = 0, i = 1, 2, …, k.
(39)
Thus, from expressions (32), (34) and (39), we get
β = ν y.
(40)
Using equations (39) and (40) in equation (33), we obtain
which further using hypothesis (iii), gives
From linearly independency of vectors {∇ y f i ( x , y ) + ∇ yy fi ( x , y ) pi }ik=1 and pi = 0, we have
αi = νλi
∀i = 1, 2, … , k .
(41)
Now, using inequalities (34) and (37) in equation (41), we get ν > 0. Therefore, equation (40) yields y ∈ C2 . Further, using equations (39)–(41) in the inequality (20), we obtain
(42) Let x ∈ C1. Then x + x ∈ C1 and hence inequality (42) implies
Also, letting x = 0 in inequality (42), we have
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Hence ( x , y , w, r = 0, s = 0) is feasible for (MSD)λ and the objective function values of (MSP) and (MSD)λ , are equal. Now, suppose that ( x , y , w, r , s ) is not an efficient solution for (MSD)λ . Then there exists (u, v, w, r, s) feasible for (MSD)λ , such that
which contradicts the weak duality theorem. Hence ( x , y , w, r = 0, s = 0) is an efficient □ solution for (MSD)λ . Theorem 4 (Converse duality): Let (u , v , λ , w, r , s ) be an efficient solution for ( MSP )λ . Suppose that (i) for i = 1, 2, …, k, ∇xxfi (u , v )
is positive definite and
or ∇xxfi (u , v ) is negative definite and
k
∑λ r
i i
T
k
∑λ r
i i
T
(∇ x f i ( x , v )) ≥ 0
i =1
(∇ x fi (u , v )) ≤ 0,
i =1
(ii) the set of vectors {∇ x fi (u , v ) + ∇ xx f i (u , v )ri }ik=1 is linearly independent, (iii) (iv)
R+k ⊆ K .
Then, ri = 0 for i = 1, 2, …, k, (u , v , w, p = 0, q = 0) is feasible for (MSP)λ , and the objective function values of (MSP)λ and (MSD) are equal. Furthermore, if the hypotheses of Theorem 1 or Theorem 2 are satisfied for all feasible solutions of (MSP)λ and (MSD), then (u , v , w, p, q ) is an efficient solution for (MSP)λ . Proof: Follows on the lines of Theorem 3. Remark 3: (i) If we take qj = q, sj = s for all j = 1, 2, …, l, then our programs become Primal (MSP′)
□
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Dual (MSD′)
Hence, the duality results for the above pair, with or without non-negativity restrictions on w (see Remark 1 or Remark 2(ii)), can also be obtained from this paper. (ii) If we let pi = 0, ri = 0, for all i = 1, 2, …, k and omitting the constraint w ≥ 0, then our programs (MSP′) and (MSD′) reduce to the problems studied in Gulati and Mehndiratta (2010b). Although in Gulati and Mehndiratta (2010b), if we take wTg = ζ (say), (where ζ: Rn × Rm → R), in the entire paper, then also the duality relations obtained in Gulati and Mehndiratta (2010b), remains same. Hence, if we ignore the non-negativity condition on w from (MSP) and (MSD), then there is no need to take w and g, separately.
5
Special cases
In this section, we consider some of the special cases of our problems studied in Section 3. For i = 1, 2, ..., k, j = 1, 2, …, l, (i) In view of Remark 3 (ii), our programs reduce to Gulati and Mehndiratta (2010b). Taking g(x, y) to be a linear function, we obtain the following particular cases: (ii) If K = R+k , C1 = R+n , and C2 = R+m , then (MSP) and (MSD) reduce to the problems studied in Suneja et al. (2003). (iii) Let pi = 0 and ri = 0 for all i then our programs become the problems considered in Khurana (2005). In addition, if K = R+k , C1 = R+n and C2 = R+m , then we get the programs studied in Weir and Mond (1988) (see (P2) and (D2) discussed in Section 1). (iv) If we set pi = 0, k = 1, K = R+ and λ = 1 ∈ int R+, then we get the programs consider in Chandra and Kumar (1998) (see (P3) and (D3) in Section 1).
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Numerical example
Let k = n = 2 and l = m = 1. Define f: R2 × R → R2 as f(x, y) = (f1(x, y), f2(x, y)), where f1(x, y) = x12 + 4x2 – y2, f2(x, y) = 3xl + x22 − y2. Let g: R2 × R → R be given by g(x, y) = x1 + 3x2 – 6y. For K = R+2 , C1 = R+2 and C2 = R+, int K* = int ( R+2 ) = {(x, y): x > 0, y > 0}, C1* = R+2 and C2* = R+ and hence our problems (MSP) and (MSD) become Primal (EMSP)
(43) (44) (45) Dual (EMSD)
(46) (47) (48) (49) 2
2
Suppose η1: S1 × S1 → R and η2: S2 × S2 → R, where S1 ⊆ R and S2 ⊆ R are given by and
η2(v, y) = (v – y), respectively. Then the hypotheses (A) and (B) of Theorem 1 reduce to x ∈ R+2 and v ∈ R+ or x1, x2 ≥ 0 and v ≥ 0. These are the primal and dual constraints (45) and (49) and hence these hypotheses are satisfied. Further
Therefore, the hypothesis (C) also holds. Next, to prove {f1(., v), f2(., v)} to be second-order K-η1-bonvex at u with K = R+2 and η1(x, u) = (x – u), we need to show
Duality for second-order symmetric multiobjective programming
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Now, let
Hence the result. Similarly, it can be shown that –{f1(x,.), f2(x,.)} is second-order K-G-convex at y. Therefore all the conditions of Theorem 2 are satisfied. Hence the result of Weak duality theorem (Theorem 2) holds. Otherwise if
This together with λ1, λ2 > 0 implies
(50) From primal constraint (44) and dual constraint (48), we have (51) and (52) Now,
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which contradicts equation (50). Hence the Theorem 1 is verified. For the dual pair (EMSP) and (EMSD), the Theorem 2 also holds for sublinear functional Fx,u(a) = (x – u)T a and Gv,y(b) = (v – y)Tb, since the hypotheses (A') and (B') of Theorem 2 reduces to xTa ≥ 0 for all a ∈ R+2 and vT b ≥ 0 for all b ∈ R+ which in turn yield x1, x2 ≥ 0 and v ≥ 0, respectively. These are the primal and dual constraints (45) and (49), hence these hypotheses hold true. Further Fx,u(∇xxg(u, v)s) = Fx,u(0) = 0 and Gv,y(–∇yyg(x, y)q) = Gv,y(0) = 0. Therefore the hypothesis (C) is satisfied. Proof of the remaining part of this theorem follows on same lines of the example verified above.
7
Conclusion and future directions
In this article, motivated by Suneja et al. (2003) and Gulati and Mehndiratta (2010b), a pair of Mond-Weir type multiobjective symmetric second-order dual programs over cones has been formulated. Further, under K-η-bonvexity/second-order K-H-convexity assumptions, weak, strong and converse duality relations are obtained. Also a nontrivial example discussed here to verify our weak duality theorems. Our study generalise some of the known work in the literature including Gulati and Mehndiratta (2010b), Khurana (2005), Suneja et al. (2003), Chandra and Kumar (1998) and Weir and Mond (1988). The present work in this paper is limited to second-order only and hence is not so efficient to find the bounds for the value of objective function of the problem if approximations are used. However this study can be further extended to the higher-order models to obtain more tighter bounds. Some of them are listed below: (i)
Higher-order symmetric multiobjective dual programs
By introducing two differentiable functions hi: Rn × Rm → R and Hj: Rn × Rm → R, for i = 1, 2, …, k and j = 1, 2, …, l our models (MSP) and (MSD) can be further extended to: Primal (HMSP)
Duality for second-order symmetric multiobjective programming
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Dual (HMSD)
One can try to obtain the duality results between (HMSP) and (HMSD) under generalised convexity assumptions. (ii) Nondifferentiable multiobjective higher-order symmetric dual programs Our programs (MSP) and (MSD) can also be extended to a following nondifferentiable problem by introducing a function called support function (defined as S(x|C) = max{xTy: y ∈ C}, where C is a compact convex set in Rn) in each of the objective functions: Primal (NMSP)
Dual (NMSD)
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where Di and Ei are compact convex sets in Rn and Rm, respectively. Further one can try to prove duality relations for the above two pair under appropriate generalised convexity assumptions. (iii) Wolfe type higher-order nondifferentiable multiobjective symmetric dual programs Including a support function, a nondifferentiable term, in each of the objective, a different model can be constructed called Wolfe type problem given below: Primal (WMP)
Dual (WMD)
and the duality theorems can be tried under (F, α, ρ, d)-convexity assumptions.
Acknowledgements The authors thank to reviewers for their constructive and helpful comments. This significantly improve the presentation of the paper. The second author is also thankful to Ministry of Human Resource development, New Delhi (India) for financial support.
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References Agarwal, R.P., Ahmad, I. and Gupta, S.K. (2011) ‘A note on higher-order non differentiable symmetric duality in multiobjective programming’, Applied Mathematics Letters, Vol. 24, pp.1308–1311. Ahmad, I. (2005a) ‘Second order symmetric duality in nondifferentiable multiobjective programming’, Information Sciences, Vol. 173, pp.23–34. Ahmad, I. (2005b) ‘Multiobjective mixed symmetric duality with invexity’, New Zealand Journal of Mathematics, Vol. 34, pp.1–9. Ahmad, I. and Husain, Z. (2005) ‘Nondifferentiable second order symmetric duality in multiobjective programming’, Applied Mathematics Letters, Vol. 18, pp.721–728. Ahmad, I. and Husain, Z. (2010a) ‘On multiobjective second order symmetric duality with cone constraints’, European Journal of Operational Research, Vol. 204, pp.402–409. Ahmad, I. and Husain, Z. (2010b) ‘Multiobjective mixed symmetric duality involving cones’, Computers and Mathematics with Applications, Vol. 59, pp.319–326. Ahmad, I., Husain, Z. and Al-Homidana, S. (2011) ‘Second-order duality in nondifferentiable fractional programming’, Nonlinear Analysis: Real World Applications, Vol. 12, pp.1103–1110. Alidaee, B., Kochenberger, G.A., Lewis, K., Lewis, M. and Wang, H. (2009) ‘Computationally attractive non-linear models for combinatorial optimization’, International Journal of Mathematics in Operational Research, Vol. 1, pp.9–19. Bazaraa, M.S. and Goode, J.J. (1973) ‘On symmetric duality in nonlinear programming’, Operations Research, Vol. 21, pp.1–9. Bector, C.R. and Chandra, S. (1985) Generalized Bonvex Functions and Second Order Duality in Mathematical Programming, Research Report 85-2, Department of Actuarial and Management Sciences, The University of Manitoba, Winnipeg, January. Bector, C.R., Chandra, S. and Goyal, A. (1999) ‘On mixed symmetric duality in multiobjective programming’, Opsearch, Vol. 36, pp.399–407. Chan, Z.X. and Sun, D. (2008) ‘Constraint nondegeneracy, strong regularity and nonsingularity in semidefinite programming’, SIAM Journal on Optimization, Vol. 19, No. 1, pp.370–396. Chandra, S. and Kumar, V. (1998) ‘A note on pseudoinvexity and symmetric duality’, European Journal of Operational Research, Vol. 105, pp.626–629. Chen, X. (2004) ‘Higher-order symmetric duality in nondifferentiable multiobjective programming problems’, Journal of Mathematical Analysis and Applications, Vol. 290, pp.423–435. Dantzig, G.B., Eisenberg, E. and Cottle, R.W. (1965) ‘Symmetric dual nolinear programming’, Pacific Journal of Mathematics, Vol. 15, pp.809–812. Dorn, W.S. (1960) ‘A symmetric dual theorem for quadratic programing’, Journal of the Operations Research Society of Japan, Vol. 2, pp.93–97. Egudo, R.R. and Mond, B. (1986) ‘Duality with generalized convexity’, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, Vol. 28, pp.10–21. Gulati, T.R. and Craven, B.D. (1983) ‘A strict converse duality theorem in nonlinear programming’, Journal of Information & Optimization Sciences, Vol. 4, pp.301–306. Gulati, T.R. and Gupta, S.K. (2005) ‘Wolfe type second-order symmetric duality in nondifferentiable programming’, J. Math. Anal. Appl., Vol. 310, pp.247–253. Gulati, T.R. and Mehndiratta, G. (2009) ‘On some symmetric dual models in multiobjective programming’, Applied Mathematics and Computation, Vol. 215, pp.380–383. Gulati, T.R. and Mehndiratta, G. (2010a) ‘Nondifferentiable multiobjective Mond-Weir type second-order symmetric duality over cones’, Optimization Letters, Vol. 4, pp.293–309. Gulati, T.R. and Mehndiratta, G. (2010b) ‘Mond-Weir type second-order symmetric duality in multiobjective programming over cones’, Applied Mathematics Letters, Vol. 23, pp.466–471.
150
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Gulati, T.R., Gupta, S.K. and Ahmad, I. (2008) ‘Second-order symmetric duality with cone constraints’, Journal of Computational and Applied Mathematics, Vol. 220, pp.347–354. Gulati, T.R., Husain, I. and Ahmed, A. (1997) ‘Multiobjective symmetric duality with invexity’, Bulletin of the Australian Mathematical Society, Vol. 56, pp.25–36. Gulati, T.R., Mehndiratta, G. and Gupta, S.K. (2011) ‘Nondifferentiable multiobjective symmetric dual programs over cones’, Nonlinear Analysis: Theory, Methods & Applications, Vol. 74, pp.3861–3869. Gulati, T.R., Saini, H. and Gupta, S.K. (2010) ‘Second-order multiobjective symmetric duality with cone constraints’, European Journal of Operational Research, Vol. 205, pp.247–252. Gupta, B. (1983) ‘Second order duality and symmetric duality for non-linear programs in complex space’, Journal of Mathematical Analysis and Applications, Vol. 97, No. 1, pp.56–64. Gupta, S.K. and Jayswal, A. (2010) ‘Multiobjective higher-order symmetric duality involving generalized cone-invex functions’, Computers and Mathematics with Applications, Vol. 60, pp.3187–3192. Gupta, S.K. and Kailey, N. (2011) ‘Nondifferentiable multiobjective second-order symmetric duality’, Optimization Letters, Vol. 5, pp.125–139. Hanson, M.A. and Mond, B. (1982) ‘Further generalizations of convexity in mathematical programming’, Journal of Information & Optimization Sciences , Vol. 3, pp.25–32. Haugh, M.B., Kogan, L. and Wang, J. (2006) ‘Evaluating portfolio policies: a duality approach’, Operations Research, Vol. 54, No. 3, pp.405–418. Jayswal, A. and Stancu-Minasian, I. (2011) ‘Higher-order duality for nondifferentiable minimax programming’, Nonlinear Analysis: Theory, Methods & Applications, Vol. 74, pp.616–625. Kassem, M.A.E-H. (2006) ‘Symmetric and self duality in vector optimization problem’, Applied Mathematics and Computation, Vol. 183, pp.1121–1126. Khurana, S. (2005) ‘Symmetric duality in multiobjective programming involving generalized cone-invex functions’, European Journal of Operational Research, Vol. 165, pp.592–597. Kim, D.S. and Lee, Y.J. (2009) ‘Nondifferentiable higher order duality in mult-iobjective programming involving cones’, Nonlinear Analysis: Theory, Methods & Applications, Vol. 71, pp.2474–2480. Kim, D.S., Yun, Y.B. and Kuk, H. (1997) ‘Second-order symmetric and self duality in multiobjective programming’, Applied Mathematics Letters, Vol. 10, No. 2, pp.17–22. Kim, M.H. and Kim, D.S. (2008) ‘Non-differentiable symmetric duality for multiobjective programming with cone constraints’, European Journal of Operational Research, Vol. 188, pp.652–661. Mangasarian, O.L. (1975) ‘Second- and higher-order duality in nonlinear programming’, Journal of Mathematical Analysis and Applications, Vol. 51, pp.607–620. Mond, B. (1974) ‘Second-order duality for nonlinear programs’, Opsearch, Vol. 11, pp.90–99. Mond, B. (1978) ‘A class of nondifferentiable fractional programming problems’, Zeitschrift fur Angewandte Mathematik und Mechanik, Vol. 58, pp.337–341. Mond, B. and Egudo, R.R. (1985) ‘On strict converse duality in nonlinear programming’, Journal of Information & Optimization Sciences, Vol. 6, pp.113–116. Mond, B. and Weir, T. (1999) ‘Generalized concavity and duality’, in Schaible, S. and Ziemba, W.T. (Eds.): Generalized Concavity in Optimization and Economics, Academic Press, New York, pp.263–280. Mordukhovich, B.S. (2007) ‘Variational analysis in nonsmooth optimization and discrete optimal control’, Mathematics of Operations Research, Vol. 32, No. 4, pp.840–856. Pandey, S. (1991) ‘Duality for multiobjective fractional programming involving generalized η-bonvex functions’, Opsearch, Vol. 28, pp.36–43.
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151
Pillo, G.D., Lucidi, S. and Palagi, L. (2005) ‘Convergence to second-order stationary points of a primal-dual algorithm model for nonlinear programming’, Mathematics of Operations Research, Vol. 30, No. 4, pp.897–915. Preda, V., Stancu-Minasian, I.M., Beldiman, M. and Stancu, A-M. (2011) ‘On a general duality model in multiobjective fractional programming with n-set functions’, Mathematical and Computer Modelling, Vol. 54, pp.490–496. Qi, H. (2009) ‘Local duality of nonlinear semidefinite programming’, Mathematics of Operations Research, Vol. 34, No. 1, pp.124–141. Saini, H. and Gulati, T.R. (2011) ‘Nondifferentiable multiobjective symmetric duality with F-convexity over cones’, Nonlinear Analysis: Theory, Methods & Applications, Vol. 74, pp.1577–1584. Schachinger, W. and Bomze, I. (2009) ‘A conic duality Frank-Wolfe-type theorem via exact penalization in quadratic optimization’, Mathematics of Operations Research, Vol. 34, No. 1, pp.83–91. Sowlati, T., Assadi, P. and Paradi, J.C. (2010) ‘Developing a mathematical programming model for senstivity analysis in analytic hierarchy process’, International Journal of Mathematics in Operational Research, Vol. 2, pp.290–301. Suneja, S.K. and Gupta, S. (1998) ‘Duality in multiobjective nonlinear programming involving semilocally convex and related functions’, European Journal of Operational Research, Vol. 107, pp.675–685. Suneja, S.K., Aggarwal, S. and Davar, S. (2002) ‘Multiobjective symmetric duality involving cones’, European Journal of Operational Research, Vol. 141, pp.471–479. Suneja, S.K., Lalitha, C.S. and Khurana, S. (2003) ‘Second-order symmetric duality in multiobjective programming’, European Journal of Operational Research, Vol. 144, pp.492–500. Weir, T. and Mond, B. (1988) ‘Symmetric and self duality in multiple objective programming’, Asia Pacific Journal of Operational Research, Vol. 5, pp.124–133. Wolfe, P. (1961) ‘A duality theorem for nonlinear programming’, Quarterly of Applied Mathematics, Vol. 19, pp.239–244. Xidonas, P., Mavrotas, G. and Psarras, J. (2010) ‘A multicriteria decision making approach for the evaluation of equity portfolios’, International Journal of Mathematics in Operational Research, Vol. 2, pp.40–72. Yang, X.M. and Hou, S.H. (2001) ‘Second-order symmetric duality in multiobjective programming’, Applied Mathematics Letters, Vol. 14, pp.587–592. Yang, X.M., Yang, X.Q., Teo, K.L. and Hou, S.H. (2005) ‘Second order symmetric duality in non-differentiable multiobjective programming with F-convexity’, European Journal of Operational Research, Vol. 164, pp.406–416. Zheng, X.Y. and Yang, X. (2007) ‘Lagrange multipliers in nonsmooth semi-infinite optimization problems’, Mathematics of Operations Research, Vol. 32, No. 1, pp.168–181.
Int. J. Mathematics in Operational Research, Vol. 4, No. 2, 2012
Duality for second-order symmetric multiobjective programming with cone constraints S.K. Gupta* and D. Dangar Department of Mathematics, Indian Institute of Technology Patna, Patna 800 013, Bihar, India Fax: +91-612-2277384 E-mail: [email protected] E-mail: [email protected] *Corresponding author Abstract: In this paper, a new pair of Mond-Weir type multiobjective second-order symmetric dual models with cone constraints is formulated in which the objective function is optimised with respect to an arbitrary closed convex cone. Usual duality relations are further established under K-η-bonvexity/second-order symmetric dual K-H-convexity assumptions. A nontrivial example has also been illustrated to justify the weak duality theorems. Several results including many recent works are obtained as special cases. Keywords: symmetric duality; K-η-bonvexity; second-order K-H-convexity; multiobjective programming; cones. Reference to this paper should be made as follows: Gupta, S.K. and Dangar, D. (2012) ‘Duality for second-order symmetric multiobjective programming with cone constraints’, Int. J. Mathematics in Operational Research, Vol. 4, No. 2, pp.128–151. Biographical notes: S.K. Gupta is an Assistant Professor in the Department of Mathematics, Indian Institute of Technology Patna, India. He obtained his MSc and PhD from Indian Institute of Technology Roorkee, India. He has published many papers in the area of Mathematical Programming. His research interests include fuzzy optimisation and support vector machines. D. Dangar recently is a Research Scholar in the Department of Mathematics, Indian Institute of Technology Patna, India. He obtained his Bachelors Degree from University of Burdwan, West Bengal, India in 2007 and Masters Degree from Banaras Hindu University, Varanasi, India in 2009.
1
Introduction
Mangasarian (1975) introduced the concept of second-order duality in nonlinear programming. Mond (1974) gave an idea of second-order convex function, which was named as bonvex function by Bector and Chandra (1985). Pandey (1991) introduced the concept of η-bonvex functions as a generalisation of bonvex functions. Yang and Hou (2001) studied a pair of second-order symmetric nondifferentiable dual programs and Copyright © 2012 Inderscience Enterprises Ltd.
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proved the duality results under F-pseudoconvexity assumptions. Alidaee et al. (2009), studied some nonlinear models and their computational significance for combinatorial optimisation problems. Wolfe and Mond-Weir type multiobjective symmetric dual programs has been discussed in Kim and Kim (2008) and proved appropriate duality relations under cone-invexity/cone-pseudoinvexity assumptions. Gulati et al. (2010) presented a pair of Wolfe and Mond-Weir type second-order symmetric multiobjective dual problems over arbitrary cones and proved weak, strong and converse duality theorems under η-bonvexity/η-pseudobonvexity assumptions. Ahmad and Husain (2010a) proved duality results under second-order invexity assumptions for a pair of Wolfe type second-order multiobjective symmetric dual programs with arbitrary cone constraints. Saini and Gulati (2011) formulated a pair of Wolfe type multiobjective second-order symmetric dual programs over arbitrary cones for nondifferentiable functions and proved duality theorems under second-order K-F-convexity assumptions. Usual duality relations has been proved in Gupta and Kailey (2011) for a pair of Wolfe type nondifferentiable second-order multiobjective dual programs under F-convexity assumptions. Gulati et al. (2011) discussed the duality results for Wolfe and Mond-Weir type nondifferentiable multiobjective symmetric dual programs over arbitrary cones under K-preinvexity/ K-convexity/pseudoinvexity assumptions. The work in Agarwal et al. (2011) fills some gap in the work of Chen (2004) by giving the correct proof of strong duality theorem for higher-order Mond-Weir type multiobjective nondifferentiable symmetric dual programs. In this paper, we consider a new pair of Mond-Weir type second-order multiobjective symmetric dual programs over arbitrary cones and proved appropriate duality relations under K-η-bonvexity/second-order K-H-convexity assumptions. We also illustrate a nontrivial example to verify our weak duality theorems.
2
Literature review
In mathematical programming, a primal-dual pair is said to be symmetric if the dual of the dual is a primal problem that is when the dual can be recast in the form of primal. Unlike linear programming, the majority of dual formulation in nonlinear programming do not possess the symmetric property. For instance, the duals formulated by Wolfe (1961) and Mond and Weir (1999) are not symmetric. Dorn (1960) first introduced the concept of symmetric duality in quadratic programming. Introducing a differentiable function ψ: Rn × Rm → R, the work in Dorn (1960) was later on extended to following nonlinear symmetric dual programs by Dantzig et al. (1965): (P1)
Minimise ψ ( x, y ) − yT ∇ yψ ( x, y ) subject to ∇ yψ ( x, y ) ≤ 0, x ≥ 0, y ≥ 0.
(D1)
Maximise ψ (u , v) − u T ∇ xψ (u, v) subject to ∇ xψ (u , v) ≥ 0, u ≥ 0, v ≥ 0.
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To weaken convexity-concavity assumptions on ψ(x, y) to pseudoconvexitypseudoconcavity, Weir and Mond (1988) studied the following pair of symmetric dual programs: (P2)
Minimise ψ ( x, y ) subject to ∇ yψ ( x, y ) ≤ 0, yT ∇ yψ ( x, y ) ≥ 0 x ≥ 0.
(D2)
Maximise ψ (u , v) subject to ∇ xψ (u , v) ≤ 0, u T ∇ xψ (u , v) ≤ 0 v ≥ 0.
Bazaraa and Goode (1973) established symmetric duality results for functions with arbitrary cones. Such a formulations enables one to consider infinitely many constraints of the inequality type. Motivated by Bazaraa and Goode (1973) and Weir and Mond (1988), Chandra and Kumar (1998) formulated the following Mond-Weir type symmetric dual programs over arbitrary cones and proved the results under pseudoinvexity type assumptions: (P3)
Minimise ψ ( x, y ) subject to ∇ yψ ( x, y ) ∈ C2* , y T ∇ yψ ( x, y ) ≥ 0, x ∈ C1 .
(D3)
Maximise ψ (u, v) subject to − ∇ xψ (u, v) ∈ C1* , u T ∇ xψ (u, v) ≤ 0, v ∈ C2 ,
where C1* and C2* are the polar cones of closed convex cones C1 and C2, respectively. Hanson and Mond (1982) formulated a type of generalised convexity and established duality relations between the general nonlinear programming problem and its Wolfe dual. Gulati and Craven (1983) and Mond and Egudo (1985) gave a strict converse duality in nonlinear programming. Egudo and Mond (1986) established duality theorems between the nonlinear programming problem and its Mond-Weir dual under F-quasiconvex/pseudoconvex assumptions and also generalise the strict converse duality theorem given in Gulati and Craven (1983) and Mond and Egudo (1985). A second-order dual for a nonlinear program with linear constraints in complex space over arbitrary polyhedral convex cones is presented and usual duality results are established in Gupta (1983). Suneja and Gupta (1998) described Wolfe and Mond-Weir type duals for a multiobjective nonlinear programming problem involving semilocally convex and related functions in terms of their right differentials and proved duality results under semilocal convexity/quasiconvexity/pseudoconvexity assumptions.
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Gulati et al. (1997) have formulated Wolfe and Mond-Weir type multiobjective symmetric dual problems and proved usual duality results under invexity/generalised invexity assumptions without the nonnegativity restrictions. Chandra and Kumar (1998) has taken a Mond-Weir type symmetric dual programs over arbitrary cones and established duality relations under pseudoinvexity assumptions. Chan and Sun (2008) reported for semidefinite programming that the primal/dual constraint non-degeneracy is equivalent to the dual/primal strong second-order sufficient condition. Following the work in Chan and Sun (2008), Qi (2009) discussed duality results in nonlinear semidefinite programming. Nonlinear optimisation problems have many applications like quadratic programming problems are used in portfolio optimisation, inventory management, engineering design etc. The performance of a given portfolio policy can be evaluated by comparing its expected utility with that of the optimal policy, which in general is not computable, in that case a direct comparison is not possible. Haugh et al. (2006) solved this problem by using the given portfolio policy to the construction based on a dual formulation of the portfolio optimisation problem. Multiobjective problems have vast number of applications in practical real life situations like multicriteria decision making approach, which is used for sensitivity analysis in analytic hierarchy process (see Sowlati et al., 2010), portfolio optimisation (see Xidonas et al., 2010), goal programming, risk programming etc. A pair of second-order Lagrangian primal-dual algorithm for inequality constrained optimisation problems that generates a sequence converging to points satisfying the second-order necessary conditions for optimality has discussed by Pillo et al. (2005). Mordukhovich (2007) devoted his study to apply the modern methods of variational analysis to constrained optimisation and control problems. Actually his main focus was on the discussion of problems with nonsmooth structures. Zheng and Yang (2007) provided Lagrange multiplier rules for a class of semi-infinite optimisation problems where all functions are lower semicontinuous using the variational analysis technique. Schachinger and Bomze (2009) described optimisation of quadratic functions over a polyhedron and applied the results to establish a Frank-Wolfe-type theorem for the primal-dual pair of a class of conic programs which shows the existence of the solution of the dual. Suneja et al. (2003) proved weak, strong and converse duality theorems under η-bonvexity/η-pseudobonvexity assumptions for the Mond-Weir type second-order symmetric multiobjective dual models without nonnegativity constraints. Ahmad and Husain (2005) and Ahmad (2005a) considered a slightly different type of Mond-Weir nondifferentiable multiobjective second-order symmetric dual programs and proved appropriate duality results under F-pseudoconvexity/F-pseudoconcavity and η-bonvexity/η-boncavity, assumptions, respectively. Gulati and Gupta (2005) studied a pair of Wolfe type second-order symmetric dual programs with nondifferentiable functions and proved duality theorems under bonvexity/boncavity assumptions. Second-order symmetric duality for nondifferentiable multiobjective programs has been taken in Ahmad (2005a) and proved duality theorems under η-pseudobonvexity assumptions. Gulati et al. (2008) proved weak, strong and converse duality theorems for Wolfe and Mond-Weir type second-order symmetric duals with differentiable functions over arbitrary cones under the η-bonvexity/ η-pseudobonvexity assumptions. A pair of Mond-Weir type multiobjective second-order symmetric duality over cones have described in Gulati and Mehndiratta (2010b) for differentiable functions proved duality relations under K-F-convexity/K-η-bonvexity
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assumptions. Motivated by Bector et al. (1999), Ahmad (2005b), Suneja et al. (2002) and Khurana (2005), Ahmad and Husain (2010b) formulate a pair of multiobjective mixed symmetric dual programs over arbitrary cones and derived duality results under cone-invexity/pseudoinvexity assumptions. Mond (1978) studied nonlinear fractional programming problems and further established necessary and sufficient conditions for optimality. Ahmad et al. (2011) presented a second-order dual for a nondifferentiable fractional programming problem which consists of maximising the ratio of functions involving square root terms of positive semidefinite quadratic forms and established duality results using second-order (F, α, ρ, d)-convexity assumptions. Gupta and Jayswal (2010) gave a Mond-Weir type higher-order multiobjective symmetric dual programs over arbitrary cones and proved duality results under higher-order cone-invexity/pseudoinvexity assumptions. Preda et al. (2011) established duality results for a fractional programming problem by replacing convexity/sublinearity assumptions on F by quasiconvexity. A class of nondifferentiable minimax programming problems has been discussed and further duality results are established in Jayswal and Stancu-Minasian (2011). Kim and Lee (2009) considered Wolfe and Mond-Weir type higher-order multiobjective dual programs over arbitrary cones and established duality relations under higher-order pseudo-type I/(F, ρ)-type I assumptions.
3
Notations and definitions
Consider the following multiobjective programming problem: (P )
K -Minimise ξ (x) subject to x ∈ X ° = {x ∈ S : −φ ( x) ∈ Q},
where S ⊆ Rn be open, ξ: S → Rk, φ: S → Rm, K is closed convex pointed cone and Q is closed convex cone with nonempty interiors in Rk and Rm, respectively. Definition 1 (Gulati and Mehndiratta, 2010b): A point x ∈ X° is an efficient solution of (P) if there exists no x ∈ X° such that
ξ ( x ) − ξ ( x) ∈ K \ {0}. Definition 2 (Chandra and Kumar, 1998): A set C of Rn is called a cone, if for each x ∈ C and λ ∈ R, λ 0 0 , we have λx ∈ C. Moreover, if C is convex then it is called convex cone. Definition 3 (Suneja et al., 2002): The positive polar cone C* of a cone C is defined by C * = {z ∈ R n : xT z ≥ 0 for all x ∈ C}.
Let T be a closed convex cone in Rn, with non-empty interior. Let S ⊆ Rn be an open set such that T ⊂ S.
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Definition 4 (Gulati and Mehndiratta, 2010a): A twice differentiable function ξ : S → Rk is said to be K-η-bonvex at u ∈ S if there exists η : S × S → Rn such that for all (x, ri) ∈ S × Rn, i = 1, 2, …, k, ξ1 ( x) − ξ1 (u ) + 1/ 2r1T ∇ xxξ1 (u )r1 − η T ( x, u )(∇ xξ1 (u ) + ∇ xxξ1 (u )r1 ),… , ∈ K . T T ξ k ( x) − ξ k (u ) + 1/ 2rk ∇ xxξ k (u )rk − η ( x, u )(∇ xξ k (u ) + ∇ xxξ k (u )rk )
Definition 5 (Ahmad and Husain, 2005): A functional H: S × S × Rn → R is said to be sublinear with respect to the third variable if for all (x, u) ∈ S × S, •
H(x, u; a1 + a2) / H(x, u; a1) + H(x, u; a2) for all a1, a2 ∈ Rn
•
H(x, u; αa) = αH(x, u; a), for all α ∈ R+ and for all a ∈ Rn.
For notational convenience, we write H(x, u; a) = Hx,u(a). Definition 6 (Gulati and Mehndiratta, 2010a): Let H be a sublinear functional with respect to the third variable and ξ : S → Rk be a twice differentiable function. Then ξ is said to be second-order K-H-convex at u ∈ S, if for all (x, ri) ∈ S × Rn, i = 1, 2, ..., k, ξ1 ( x) − ξ1 (u ) + 1/ 2r1T ∇ xxξ1 (u )r1 − H x ,u (∇ xξ1 (u ) + ∇ xxξ1 (u )r1 ),… , ∈ K. ξ ( x) − ξ (u ) + 1/ 2r T ∇ ξ (u )r − H (∇ ξ (u ) + ∇ ξ (u )r ) k k xx k k x ,u x k xx k k k
Throughout this paper, we have taken p = (p1, …, pk), q = (q1, …, ql), r = (r1, …, rk), s = (s1, …, sl), λ = (λ1, …, λk) and w = (w1, …, wl), where pi, qj ∈ Rm, ri, sj ∈ Rn, λi, wj ∈ R for i = 1, 2, …, k, j = 1, 2, …, l.
4
Second-order symmetric duality
Suneja et al. (2002), formulated a pair of multiobjective symmetric dual programs over arbitrary cones and established weak, strong, converse and self duality theorems under K-convexity/K-concavity assumptions. Khurana (2005) proved duality theorems under K-pseudoinvexity assumptions for a pair of Mond-Weir type dual programs. Yang et al. (2005) established duality relations under F-convexity assumptions for the Mond-Weir type second-order symmetric nondifferentiable multiobjective dual problems. Kassem (2006) presented a pair of symmetric multiobjective dual programs and proved duality relations under convexity/concavity assumptions. Gulati and Mehndiratta (2009) pointed out certain omissions in Kaseem (2006) and Kim et al. (1997) and gave corrected models and proofs of their duality theorems. Recently Gulati et al. (2010) formulated Wolfe and Mond-Weir type second-order multiobjective symmetric dual for differentiable functions and Gulati and Mehndiratta (2010a) considered for nondifferentiable case. Further, they proved duality theorems under η-bonvexity/η-pseudobonvexity and K-η-bonvexity/ K-F-convexity assumptions, respectively. Inspired by above mentioned papers, we formulate the following pair of Mond-Weir type second-order multiobjective symmetric dual programs over arbitrary cones:
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Primal (MSP) K - mi nimise L( x, y, λ , w, p, q ) = { f1 ( x, y ) − 1/ 2 p1T ∇ yy f1 ( x, y ) p1 ,… , f k ( x, y ) − 1/ 2 pkT ∇ yy f k ( x, y ) pk }
subject to l k − ∑ λi ( ∇ y fi ( x, y ) + ∇ yy fi ( x, y ) pi ) + ∑ w j ∇ yy g j ( x, y )q j ∈ C2* , j =1 i =1
(1)
l k yT ∑ λi ( ∇ y f i ( x, y ) + ∇ yy f i ( x, y ) pi ) + ∑ w j ∇ yy g j ( x, y )q j ≥ 0, j =1 i =1
(2)
λ ∈ int K * , x ∈ C1 , w j ≥ 0, j = 1, 2, … , l ,
(3)
Dual (MSD) K -maximise M (u , v, λ , w, r , s ) = { f1 (u, v) − 1/ 2r1T ∇ xx f1 (u , v)r1 ,… , f k (u , v) − 1/ 2rkT ∇ xx f k (u , v)rk }
subject to k
∑ λ (∇ i
i =1
l
x
fi (u, v) + ∇ xx fi (u , v)ri ) + ∑ w j ∇ xx g j (u, v) s j ∈ C1* ,
(4)
j =1
l k u T ∑ λi ( ∇ x f i (u , v) + ∇ xx f i (u, v)ri ) + ∑ w j ∇ xx g j (u , v) s j ≤ 0, j =1 i =1
(5)
λ ∈ int K * , v ∈ C2 , w j ≥ 0, j = 1, 2, … , l ,
(6)
where for i = 1, 2, …, k, j = 1, 2, …,l, •
C1 and C2 are closed convex cone with non-empty interior in Rn and Rm, respectively,
•
S1 ⊆ Rn and S2 ⊆ Rm are open sets such that C1 × C2 ⊂ S1 × S2,
•
fi : S1 × S2 → R and gj : S1 × S2 → R, are twice differentiable functions of x and y
•
C1* , C2* and K * are positive polar cones of Cl, C2 and K, respectively.
Now, we prove the weak, strong, converse duality theorems for dual pair (MSP) and (MSD). Theorem 1 (Weak duality): Let (x, y, λ, w, p, q) and (u, v, λ, w, r, s) be feasible for (MSP) and (MSD), respectively. Suppose there exist functions η1 : Sl × Sl → Rn and η2 : S2 × S2 → Rm satisfying
η1 ( x, u ) + u ∈ C1 ,
(A)
η 2 ( v, y ) + y ∈ C2 ,
(B)
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0 ∇ g (u , v) sj 0 (η1T ( x, u ) η 2T (v, y )) xx j ≤ 0, −∇ yy g j ( x, y ) 0 q j 0 for j = 1,2, … , l.
(C)
Furthermore, let {f1(., v), …, fk (., v)} be K-η1-bonvex at u and –{f1(x,.), …, fk(x,.)} be K-η2-bonvex at y. Then L( x, y, λ , w, p, q) − M (u , v, λ , w, r , s ) ∉ − K \ {0}.
(7)
Proof: By the K-η1-bonvexity of {f1(., v), …, fk(., v)} at u, we have ( f1 ( x, v) − f1 (u , v) + 1/ 2r1T ∇ xx f1 (u , v )r1 − η1T ( x, u )(∇ x f1 (u , v ) + ∇ xx f1 (u , v)r1 ),… , f k ( x, v) − f k (u, v) + 1/ 2rkT ∇ xx f k (u , v )rk − η1T ( x, u )(∇ x f k (u, v) + ∇ xx f k (u , v) rk ) ∈ K .
As λ ∈ int K*, therefore from above we obtain k
∑ λ ( f ( x, v) − f (u, v) + 1/ 2r i
i =1
i
i
T
i
∇ xx f i (u, v)ri
(8)
−η1T ( x, u )(∇ x fi (u , v) + ∇ xx f i (u, v)ri ) ) ≥ 0.
Now, it follows from (A) and dual constraint (4) that
k
l
j =1
(η1 ( x, u ) + u ) ∑ λi ( ∇ x fi (u, v) + ∇ xx fi (u, v)ri ) + ∑ w j ∇ xx g j (u, v) s j ≥ 0, T
i =1
which using inequality (5) implies
k
l
j =1
η1T ( x, u ) ∑ λi ( ∇ x f i (u , v) + ∇ xx f i (u, v)ri ) + ∑ w j ∇ xx g j (u , v) s j ≥ 0. i =1
This further by hypothesis (C) and w j ≥ 0, j = 1, 2, … , l , yields
k
i =1
η1T ( x, u ) ∑ λi ( ∇ x f i (u , v) + ∇ xx fi (u , v)ri ) ≥ 0.
(9)
From inequalities (8) and (9), we get k
∑ λ ( f ( x, v) − f (u, v) + 1/ 2r i
i
i
i =1
i
T
∇ xx f i (u, v)ri ) ≥ 0.
(10)
Similarly, using K-η2-bonvexity of –{f1(x,.), …, fk(x,.)} at y, hypotheses (B), (C) and primal constraints (1) and (2), we obtain k
∑ λ ( f ( x, y) − f ( x, v) − 1/ 2 p ∇ i
i =1
i
i
T i
yy
fi ( x, y ) pi ) ≥ 0.
(11)
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Adding inequalities (10) and (11), we have k
∑ λ ( f ( x, y) − 1/ 2 p i
T i
i
i =1
∇ yy f i ( x, y ) pi − fi (u , v) + 1/ 2riT ∇ xx f i (u, v) ri ) ≥ 0.
(12)
But suppose that L( x, y, λ , w, p, q ) − M (u, v, λ , w, r , s ) ∈ − K \ {0}. i.e.
{( f ( x, y) − 1/ 2 p ∇ T 1
1
f ( x, y ) p1 , …, f k ( x, y ) − 1/ 2 pkT ∇ yy f k ( x, y ) pk )
yy 1
}
−( f1 (u, v) − 1/ 2r1T ∇ xx f1 (u , v)r1 , … , f k (u , v ) − 1/ 2rkT ∇ xx f k (u, v)rk ) ∈ − K \ {0}.
Since λ ∈ int K*, it yields k
∑ λ ( ( f ( x, y) − 1/ 2 p i
i
i =1
T i
∇ yy f i ( x, y ) pi ) − ( fi (u, v) − 1/ 2riT ∇ xx f i (u, v) ri ) ) < 0,
which contradicts inequality (12). Hence the result.
□
Remark 1: If we replace the hypothesis (C) in the above theorem by 0 l ∑ w j ∇ xx g j (u, v) s j l T T (η1 ( x, u ) η 2 (v, y )) j =1 ≤ 0, − ∇ w g x y q ( , ) ∑ j yy j j 0 j =1
then the same conclusion of Theorem 1 also holds. However, taking this hypothesis, the theorem can be obtained without even considering the non-negativity restriction on wj, j = 1, 2, …, l. Theorem 2 (Weak duality): Let (x, y, λ, w, p, q) and (u, v, λ, w, r, s) be feasible for (MSP) and (MSD), respectively. Let for sublinear functionals F: S1 × S1 × Rn → R and G: S2 × S2 × Rm → R satisfying Fx ,u (a ) + u T a ≥ 0
for all a ∈ C1* ,
(A′)
Gv , y (b) + y T b ≥ 0
for all b ∈ C2* ,
(B′)
Fx ,u (∇ xx g j (u , v ) s j ) ≤ 0 and Gv , y (−∇ yy g j ( x, y )q j ) ≤ 0,
for j = 1, 2, …, l.
(C′)
Furthermore, let {f1(., v), …, fk(., v)} be second-order K-F-convex at u, –{f1(x,.), …, fk(x,.)} be second-order K-G-convex at y and R+k ⊆ K . Then L( x, y, λ , w, p, q ) − M (u, v, λ , w, r , s ) ∉ − K \ {0}.
Proof: By second-order K-F-convexity of {f1(., v), …, fk(., v)} at u, we have ( f1 ( x, v) − f1 (u , v) + 1/ 2r1T ∇ xx f1 (u , v)r1 − Fx ,u (∇ x f1 (u, v) + ∇ xx f1 (u, v) r1 ),… , f k ( x, v) − f k (u, v) + 1/ 2rkT ∇ xx f k (u , v)rk − Fx ,u (∇ x f k (u , v) + ∇ xx f k (u, v)rk )) ∈ K .
(13)
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This together with λ ∈ int K* implies k
∑ λ ( f ( x, v) − f (u, v) + 1/ 2r i
i
i
T
i
i =1
∇ xx f i (u, v)ri )
k
≥ ∑ λi Fx ,u ( ∇ x fi (u , v) + ∇ xx f i (u , v)ri ).
(14)
i =1
As R+k ⊆ K ⇒ K * ⊆ R+k and since λ ∈ int K*, therefore
λ > 0.
(15)
Using inequality (15) and sublinearity of F in inequality (14), we have k
∑ λ ( f ( x, v) − f (u, v) + 1/ 2r i
i
i
T
i
i =1
∇ xx f i (u , v)ri )
k ≥ Fx ,u ∑ λi ( ∇ x fi (u, v) + ∇ xx fi (u , v)ri ) . i =1
(16)
Further, it follows from hypothesis (A′) and inequality (4) that l k Fx ,u ∑ λi ( ∇ x fi (u, v) + ∇ xx f i (u , v)ri ) + ∑ w j ∇ xx g j (u, v) s j j =1 i =1 l k +u T ∑ λi ( ∇ x fi (u , v) + ∇ xx f i (u, v)ri ) + ∑ w j ∇ xx g j (u , v ) s j ≥ 0, j =1 i =1 k
l
i =1
j =1
for a = ∑ λi ( ∇ x f i (u, v) + ∇ xx f i (u , v)ri ) + ∑ w j ∇ xx g j (u, v) s j ∈ C1*
which by using inequality (5) implies l k Fx ,u ∑ λi ( ∇ x fi (u , v) + ∇ xx f i (u, v)ri ) + ∑ w j ∇ xx g j (u , v) s j ≥ 0. j =1 i =1
Now, applying sublinearity of F and using w j ≥ 0, for all j, we have k l Fx ,u ∑ λi (∇ x f i (u, v) + ∇ xx fi (u, v)ri ) + ∑ w j Fx ,u (∇ xx g j (u , v) s j ) ≥ 0. i =1 j =1
This using inequality (C′) and w j ≥ 0, j = 1, 2, …, l , we obtain k Fx ,u ∑ λi (∇ x f i (u, v) + ∇ xx fi (u, v)ri ) ≥ 0. i =1
It follows from inequality (16) that k
∑ λ ( f ( x, v) − f (u, v) + 1/ 2r i
i =1
i
i
i
T
∇ xx fi (u , v)ri ) ≥ 0.
(17)
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Similarly, using K-G-convexity of –{f1(x,.), …, fk(x,.)} at y, hypotheses (B′), (C′), (15), primal constraints (1) and (2), and the sublinearity of G, we obtain k
∑ λ ( f ( x, y) − f ( x, y) − 1/ 2 p i
i
T i
i
i =1
∇ yy fi ( x, y ) pi ) ≥ 0.
(18)
Finally adding inequalities (17) and (18), we have k
∑ λ ( f ( x, y) − 1/ 2 p i
T i
i
i =1
∇ yy fi ( x, y ) pi − fi (u , v) + 1/ 2riT ∇ xx f i (u , v)ri ) ≥ 0.
(19)
Now, suppose on the contrary L( x, y, λ , w, p, q ) − M (u, v, λ , w, r , s ) ∈ − K \ {0}. or
{( f ( x, y) − 1/ 2 p ∇ − { f (u , v) − 1/ 2r ∇ 1
1
T 1
yy 1
f ( x, y ) p1 , … , f k ( x, y ) − 1/ 2 pkT ∇ yy f k ( x, y ) pk
T 1
xx 1
}
}
f (u, v)r1 , … , f k (u, v) − 1/ 2rkT ∇ xx f k (u, v)rk ) ∈ − K \ {0}.
Since λ ∈ int K*, therefore we have k
∑ λ ( ( f ( x, y) − 1/ 2 p i
i =1
i
T i
∇ yy f i ( x, y ) pi ) − ( f i (u , v) − 1/ 2riT ∇ xx fi (u , v)ri ) ) < 0,
which contradicts inequality (19). Hence the result.
□
Remark 2: (i) It may be noted that to prove Theorem 2, we also need an additional assumption R+k ⊆ K . (ii) If we replace the hypothesis (C′) in the above theorem by l l Fx ,u ∑ w j ∇ xx g j (u , v) s j ≤ 0 and Gv , y −∑ w j ∇ yy g j ( x, y )q j ≤ 0. j =1 j =1
then the same conclusion of Theorem 2 also holds. Using this assumption in the proof, the requirement of non-negativity restriction on w may be removed. The notations (MSP)λ and (MSD)λ are used in the following theorems to denote (MSP) and (MSD), respectively when λ is fixed to be λ . Theorem 3 (Strong duality): Let ( x , y , λ , w, p, q ) be an efficient solution for ( MSD)λ . Let (i) for i = 1, 2, …, k, ∇ yy f i ( x , y ) be positive definite and
k
∑λ p i
T i
∇ y fi ( x , y ) ≥ 0
j =1
or ∇ yy f i ( x , y ) be negative definite and
k
∑λ p i
T i
∇ y fi ( x , y ) ≤ 0,
j =1
(ii)
the set of vectors {∇yfi ( x , y ) + ∇yyfi ( x , y ) pi }ik=1 be linearly independent,
Duality for second-order symmetric multiobjective programming (iii)
l
∑w ∇ j
yy
j =1
(iv)
139
g j ( x , y )q j ∉ span{∇ y fi ( x , y ) + ∇ yy fi ( x , y ) pi }ik=1 \{0} and
R+k ⊆ K .
Then, pi = 0 for i = 1, 2, ..., k, ( x , y , w, r = 0, s = 0) is feasible for (MSD)λ and the objective function values of (MSP) and (MSD)λ are equal. Furthermore, if the hypotheses of Theorem 1 or Theorem 2 are satisfied for all feasible solutions of (MSP) and (MSD)λ , then ( x , y , w, r , s ) is an efficient solution for (MSD)λ . Proof: Since ( x , y , λ , w, p, q ) is an efficient solution to (MSP), by Fritz John necessary optimality conditions Suneja et al. (2002), there exist α ∈ K*, β ∈ C2, v ∈ R+, δ ∈ R+l , such that the following conditions are satisfied at ( x , y , λ , w, p, q ) (for simplicity, we write ∇xfi, ∇yxfi, ∇yygj instead of ∇ x fi ( x , y ), ∇ yx fi ( x , y ), ∇ yy g j ( x , y ), etc:
(20)
(21) (22) (23) (24) (25)
(26)
(27) (28) (29)
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Inequalities (21) and (22) are equivalent to
(30) and (31) Since ∇yyfi is positive or negative definite, for i = 1, 2, …, k, equation (25) yields (32) From equations (30) and (32), we have
(33) Also, R ⊆ K ⇒ K * ⊆ R and since λ ∈ int K *, therefore k +
k +
λ > 0.
(34)
Now, we claim that αi ≠ 0 for all i = 1, 2, …, k. Otherwise if, αi = 0 for some i, say t0, 1 ≤ t0 ≤ k, then for i = t0, using inequality (34) in equation (32), we get
β = ν y.
(35)
From equations (33) and (35), we have
It follows from hypothesis (iii) that
This using hypothesis (ii), yields
αi = νλi , i = 1, 2, …, k.
(36)
Since λ > 0 and α t0 = 0, therefore from equation (36), ν = 0. Hence equations (35) and (36) implies β = 0 and αi = 0 for all i. Also, from equation (23), δj = 0, j = 1, 2, …, l. Therefore (α, β, v, δ) = 0, which contradicts (29). Hence αi ≠ 0 for all i. Moreover, α ∈ K * ⊆ R+k (by hypothesis (iv)), therefore for i = 1, 2, …, k
αi > 0.
(37)
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Further, using expressions (32) and (37) in equation (31), we have
(38) We now prove that pi = 0, i = 1, 2, …, k. Otherwise, hypothesis (i) implies
which contradicts equation (38) and hence pi = 0, i = 1, 2, …, k.
(39)
Thus, from expressions (32), (34) and (39), we get
β = ν y.
(40)
Using equations (39) and (40) in equation (33), we obtain
which further using hypothesis (iii), gives
From linearly independency of vectors {∇ y f i ( x , y ) + ∇ yy fi ( x , y ) pi }ik=1 and pi = 0, we have
αi = νλi
∀i = 1, 2, … , k .
(41)
Now, using inequalities (34) and (37) in equation (41), we get ν > 0. Therefore, equation (40) yields y ∈ C2 . Further, using equations (39)–(41) in the inequality (20), we obtain
(42) Let x ∈ C1. Then x + x ∈ C1 and hence inequality (42) implies
Also, letting x = 0 in inequality (42), we have
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Hence ( x , y , w, r = 0, s = 0) is feasible for (MSD)λ and the objective function values of (MSP) and (MSD)λ , are equal. Now, suppose that ( x , y , w, r , s ) is not an efficient solution for (MSD)λ . Then there exists (u, v, w, r, s) feasible for (MSD)λ , such that
which contradicts the weak duality theorem. Hence ( x , y , w, r = 0, s = 0) is an efficient □ solution for (MSD)λ . Theorem 4 (Converse duality): Let (u , v , λ , w, r , s ) be an efficient solution for ( MSP )λ . Suppose that (i) for i = 1, 2, …, k, ∇xxfi (u , v )
is positive definite and
or ∇xxfi (u , v ) is negative definite and
k
∑λ r
i i
T
k
∑λ r
i i
T
(∇ x f i ( x , v )) ≥ 0
i =1
(∇ x fi (u , v )) ≤ 0,
i =1
(ii) the set of vectors {∇ x fi (u , v ) + ∇ xx f i (u , v )ri }ik=1 is linearly independent, (iii) (iv)
R+k ⊆ K .
Then, ri = 0 for i = 1, 2, …, k, (u , v , w, p = 0, q = 0) is feasible for (MSP)λ , and the objective function values of (MSP)λ and (MSD) are equal. Furthermore, if the hypotheses of Theorem 1 or Theorem 2 are satisfied for all feasible solutions of (MSP)λ and (MSD), then (u , v , w, p, q ) is an efficient solution for (MSP)λ . Proof: Follows on the lines of Theorem 3. Remark 3: (i) If we take qj = q, sj = s for all j = 1, 2, …, l, then our programs become Primal (MSP′)
□
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Dual (MSD′)
Hence, the duality results for the above pair, with or without non-negativity restrictions on w (see Remark 1 or Remark 2(ii)), can also be obtained from this paper. (ii) If we let pi = 0, ri = 0, for all i = 1, 2, …, k and omitting the constraint w ≥ 0, then our programs (MSP′) and (MSD′) reduce to the problems studied in Gulati and Mehndiratta (2010b). Although in Gulati and Mehndiratta (2010b), if we take wTg = ζ (say), (where ζ: Rn × Rm → R), in the entire paper, then also the duality relations obtained in Gulati and Mehndiratta (2010b), remains same. Hence, if we ignore the non-negativity condition on w from (MSP) and (MSD), then there is no need to take w and g, separately.
5
Special cases
In this section, we consider some of the special cases of our problems studied in Section 3. For i = 1, 2, ..., k, j = 1, 2, …, l, (i) In view of Remark 3 (ii), our programs reduce to Gulati and Mehndiratta (2010b). Taking g(x, y) to be a linear function, we obtain the following particular cases: (ii) If K = R+k , C1 = R+n , and C2 = R+m , then (MSP) and (MSD) reduce to the problems studied in Suneja et al. (2003). (iii) Let pi = 0 and ri = 0 for all i then our programs become the problems considered in Khurana (2005). In addition, if K = R+k , C1 = R+n and C2 = R+m , then we get the programs studied in Weir and Mond (1988) (see (P2) and (D2) discussed in Section 1). (iv) If we set pi = 0, k = 1, K = R+ and λ = 1 ∈ int R+, then we get the programs consider in Chandra and Kumar (1998) (see (P3) and (D3) in Section 1).
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Numerical example
Let k = n = 2 and l = m = 1. Define f: R2 × R → R2 as f(x, y) = (f1(x, y), f2(x, y)), where f1(x, y) = x12 + 4x2 – y2, f2(x, y) = 3xl + x22 − y2. Let g: R2 × R → R be given by g(x, y) = x1 + 3x2 – 6y. For K = R+2 , C1 = R+2 and C2 = R+, int K* = int ( R+2 ) = {(x, y): x > 0, y > 0}, C1* = R+2 and C2* = R+ and hence our problems (MSP) and (MSD) become Primal (EMSP)
(43) (44) (45) Dual (EMSD)
(46) (47) (48) (49) 2
2
Suppose η1: S1 × S1 → R and η2: S2 × S2 → R, where S1 ⊆ R and S2 ⊆ R are given by and
η2(v, y) = (v – y), respectively. Then the hypotheses (A) and (B) of Theorem 1 reduce to x ∈ R+2 and v ∈ R+ or x1, x2 ≥ 0 and v ≥ 0. These are the primal and dual constraints (45) and (49) and hence these hypotheses are satisfied. Further
Therefore, the hypothesis (C) also holds. Next, to prove {f1(., v), f2(., v)} to be second-order K-η1-bonvex at u with K = R+2 and η1(x, u) = (x – u), we need to show
Duality for second-order symmetric multiobjective programming
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Now, let
Hence the result. Similarly, it can be shown that –{f1(x,.), f2(x,.)} is second-order K-G-convex at y. Therefore all the conditions of Theorem 2 are satisfied. Hence the result of Weak duality theorem (Theorem 2) holds. Otherwise if
This together with λ1, λ2 > 0 implies
(50) From primal constraint (44) and dual constraint (48), we have (51) and (52) Now,
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which contradicts equation (50). Hence the Theorem 1 is verified. For the dual pair (EMSP) and (EMSD), the Theorem 2 also holds for sublinear functional Fx,u(a) = (x – u)T a and Gv,y(b) = (v – y)Tb, since the hypotheses (A') and (B') of Theorem 2 reduces to xTa ≥ 0 for all a ∈ R+2 and vT b ≥ 0 for all b ∈ R+ which in turn yield x1, x2 ≥ 0 and v ≥ 0, respectively. These are the primal and dual constraints (45) and (49), hence these hypotheses hold true. Further Fx,u(∇xxg(u, v)s) = Fx,u(0) = 0 and Gv,y(–∇yyg(x, y)q) = Gv,y(0) = 0. Therefore the hypothesis (C) is satisfied. Proof of the remaining part of this theorem follows on same lines of the example verified above.
7
Conclusion and future directions
In this article, motivated by Suneja et al. (2003) and Gulati and Mehndiratta (2010b), a pair of Mond-Weir type multiobjective symmetric second-order dual programs over cones has been formulated. Further, under K-η-bonvexity/second-order K-H-convexity assumptions, weak, strong and converse duality relations are obtained. Also a nontrivial example discussed here to verify our weak duality theorems. Our study generalise some of the known work in the literature including Gulati and Mehndiratta (2010b), Khurana (2005), Suneja et al. (2003), Chandra and Kumar (1998) and Weir and Mond (1988). The present work in this paper is limited to second-order only and hence is not so efficient to find the bounds for the value of objective function of the problem if approximations are used. However this study can be further extended to the higher-order models to obtain more tighter bounds. Some of them are listed below: (i)
Higher-order symmetric multiobjective dual programs
By introducing two differentiable functions hi: Rn × Rm → R and Hj: Rn × Rm → R, for i = 1, 2, …, k and j = 1, 2, …, l our models (MSP) and (MSD) can be further extended to: Primal (HMSP)
Duality for second-order symmetric multiobjective programming
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Dual (HMSD)
One can try to obtain the duality results between (HMSP) and (HMSD) under generalised convexity assumptions. (ii) Nondifferentiable multiobjective higher-order symmetric dual programs Our programs (MSP) and (MSD) can also be extended to a following nondifferentiable problem by introducing a function called support function (defined as S(x|C) = max{xTy: y ∈ C}, where C is a compact convex set in Rn) in each of the objective functions: Primal (NMSP)
Dual (NMSD)
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where Di and Ei are compact convex sets in Rn and Rm, respectively. Further one can try to prove duality relations for the above two pair under appropriate generalised convexity assumptions. (iii) Wolfe type higher-order nondifferentiable multiobjective symmetric dual programs Including a support function, a nondifferentiable term, in each of the objective, a different model can be constructed called Wolfe type problem given below: Primal (WMP)
Dual (WMD)
and the duality theorems can be tried under (F, α, ρ, d)-convexity assumptions.
Acknowledgements The authors thank to reviewers for their constructive and helpful comments. This significantly improve the presentation of the paper. The second author is also thankful to Ministry of Human Resource development, New Delhi (India) for financial support.
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References Agarwal, R.P., Ahmad, I. and Gupta, S.K. (2011) ‘A note on higher-order non differentiable symmetric duality in multiobjective programming’, Applied Mathematics Letters, Vol. 24, pp.1308–1311. Ahmad, I. (2005a) ‘Second order symmetric duality in nondifferentiable multiobjective programming’, Information Sciences, Vol. 173, pp.23–34. Ahmad, I. (2005b) ‘Multiobjective mixed symmetric duality with invexity’, New Zealand Journal of Mathematics, Vol. 34, pp.1–9. Ahmad, I. and Husain, Z. (2005) ‘Nondifferentiable second order symmetric duality in multiobjective programming’, Applied Mathematics Letters, Vol. 18, pp.721–728. Ahmad, I. and Husain, Z. (2010a) ‘On multiobjective second order symmetric duality with cone constraints’, European Journal of Operational Research, Vol. 204, pp.402–409. Ahmad, I. and Husain, Z. (2010b) ‘Multiobjective mixed symmetric duality involving cones’, Computers and Mathematics with Applications, Vol. 59, pp.319–326. Ahmad, I., Husain, Z. and Al-Homidana, S. (2011) ‘Second-order duality in nondifferentiable fractional programming’, Nonlinear Analysis: Real World Applications, Vol. 12, pp.1103–1110. Alidaee, B., Kochenberger, G.A., Lewis, K., Lewis, M. and Wang, H. (2009) ‘Computationally attractive non-linear models for combinatorial optimization’, International Journal of Mathematics in Operational Research, Vol. 1, pp.9–19. Bazaraa, M.S. and Goode, J.J. (1973) ‘On symmetric duality in nonlinear programming’, Operations Research, Vol. 21, pp.1–9. Bector, C.R. and Chandra, S. (1985) Generalized Bonvex Functions and Second Order Duality in Mathematical Programming, Research Report 85-2, Department of Actuarial and Management Sciences, The University of Manitoba, Winnipeg, January. Bector, C.R., Chandra, S. and Goyal, A. (1999) ‘On mixed symmetric duality in multiobjective programming’, Opsearch, Vol. 36, pp.399–407. Chan, Z.X. and Sun, D. (2008) ‘Constraint nondegeneracy, strong regularity and nonsingularity in semidefinite programming’, SIAM Journal on Optimization, Vol. 19, No. 1, pp.370–396. Chandra, S. and Kumar, V. (1998) ‘A note on pseudoinvexity and symmetric duality’, European Journal of Operational Research, Vol. 105, pp.626–629. Chen, X. (2004) ‘Higher-order symmetric duality in nondifferentiable multiobjective programming problems’, Journal of Mathematical Analysis and Applications, Vol. 290, pp.423–435. Dantzig, G.B., Eisenberg, E. and Cottle, R.W. (1965) ‘Symmetric dual nolinear programming’, Pacific Journal of Mathematics, Vol. 15, pp.809–812. Dorn, W.S. (1960) ‘A symmetric dual theorem for quadratic programing’, Journal of the Operations Research Society of Japan, Vol. 2, pp.93–97. Egudo, R.R. and Mond, B. (1986) ‘Duality with generalized convexity’, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, Vol. 28, pp.10–21. Gulati, T.R. and Craven, B.D. (1983) ‘A strict converse duality theorem in nonlinear programming’, Journal of Information & Optimization Sciences, Vol. 4, pp.301–306. Gulati, T.R. and Gupta, S.K. (2005) ‘Wolfe type second-order symmetric duality in nondifferentiable programming’, J. Math. Anal. Appl., Vol. 310, pp.247–253. Gulati, T.R. and Mehndiratta, G. (2009) ‘On some symmetric dual models in multiobjective programming’, Applied Mathematics and Computation, Vol. 215, pp.380–383. Gulati, T.R. and Mehndiratta, G. (2010a) ‘Nondifferentiable multiobjective Mond-Weir type second-order symmetric duality over cones’, Optimization Letters, Vol. 4, pp.293–309. Gulati, T.R. and Mehndiratta, G. (2010b) ‘Mond-Weir type second-order symmetric duality in multiobjective programming over cones’, Applied Mathematics Letters, Vol. 23, pp.466–471.
150
S.K. Gupta and D. Dangar
Gulati, T.R., Gupta, S.K. and Ahmad, I. (2008) ‘Second-order symmetric duality with cone constraints’, Journal of Computational and Applied Mathematics, Vol. 220, pp.347–354. Gulati, T.R., Husain, I. and Ahmed, A. (1997) ‘Multiobjective symmetric duality with invexity’, Bulletin of the Australian Mathematical Society, Vol. 56, pp.25–36. Gulati, T.R., Mehndiratta, G. and Gupta, S.K. (2011) ‘Nondifferentiable multiobjective symmetric dual programs over cones’, Nonlinear Analysis: Theory, Methods & Applications, Vol. 74, pp.3861–3869. Gulati, T.R., Saini, H. and Gupta, S.K. (2010) ‘Second-order multiobjective symmetric duality with cone constraints’, European Journal of Operational Research, Vol. 205, pp.247–252. Gupta, B. (1983) ‘Second order duality and symmetric duality for non-linear programs in complex space’, Journal of Mathematical Analysis and Applications, Vol. 97, No. 1, pp.56–64. Gupta, S.K. and Jayswal, A. (2010) ‘Multiobjective higher-order symmetric duality involving generalized cone-invex functions’, Computers and Mathematics with Applications, Vol. 60, pp.3187–3192. Gupta, S.K. and Kailey, N. (2011) ‘Nondifferentiable multiobjective second-order symmetric duality’, Optimization Letters, Vol. 5, pp.125–139. Hanson, M.A. and Mond, B. (1982) ‘Further generalizations of convexity in mathematical programming’, Journal of Information & Optimization Sciences , Vol. 3, pp.25–32. Haugh, M.B., Kogan, L. and Wang, J. (2006) ‘Evaluating portfolio policies: a duality approach’, Operations Research, Vol. 54, No. 3, pp.405–418. Jayswal, A. and Stancu-Minasian, I. (2011) ‘Higher-order duality for nondifferentiable minimax programming’, Nonlinear Analysis: Theory, Methods & Applications, Vol. 74, pp.616–625. Kassem, M.A.E-H. (2006) ‘Symmetric and self duality in vector optimization problem’, Applied Mathematics and Computation, Vol. 183, pp.1121–1126. Khurana, S. (2005) ‘Symmetric duality in multiobjective programming involving generalized cone-invex functions’, European Journal of Operational Research, Vol. 165, pp.592–597. Kim, D.S. and Lee, Y.J. (2009) ‘Nondifferentiable higher order duality in mult-iobjective programming involving cones’, Nonlinear Analysis: Theory, Methods & Applications, Vol. 71, pp.2474–2480. Kim, D.S., Yun, Y.B. and Kuk, H. (1997) ‘Second-order symmetric and self duality in multiobjective programming’, Applied Mathematics Letters, Vol. 10, No. 2, pp.17–22. Kim, M.H. and Kim, D.S. (2008) ‘Non-differentiable symmetric duality for multiobjective programming with cone constraints’, European Journal of Operational Research, Vol. 188, pp.652–661. Mangasarian, O.L. (1975) ‘Second- and higher-order duality in nonlinear programming’, Journal of Mathematical Analysis and Applications, Vol. 51, pp.607–620. Mond, B. (1974) ‘Second-order duality for nonlinear programs’, Opsearch, Vol. 11, pp.90–99. Mond, B. (1978) ‘A class of nondifferentiable fractional programming problems’, Zeitschrift fur Angewandte Mathematik und Mechanik, Vol. 58, pp.337–341. Mond, B. and Egudo, R.R. (1985) ‘On strict converse duality in nonlinear programming’, Journal of Information & Optimization Sciences, Vol. 6, pp.113–116. Mond, B. and Weir, T. (1999) ‘Generalized concavity and duality’, in Schaible, S. and Ziemba, W.T. (Eds.): Generalized Concavity in Optimization and Economics, Academic Press, New York, pp.263–280. Mordukhovich, B.S. (2007) ‘Variational analysis in nonsmooth optimization and discrete optimal control’, Mathematics of Operations Research, Vol. 32, No. 4, pp.840–856. Pandey, S. (1991) ‘Duality for multiobjective fractional programming involving generalized η-bonvex functions’, Opsearch, Vol. 28, pp.36–43.
Duality for second-order symmetric multiobjective programming
151
Pillo, G.D., Lucidi, S. and Palagi, L. (2005) ‘Convergence to second-order stationary points of a primal-dual algorithm model for nonlinear programming’, Mathematics of Operations Research, Vol. 30, No. 4, pp.897–915. Preda, V., Stancu-Minasian, I.M., Beldiman, M. and Stancu, A-M. (2011) ‘On a general duality model in multiobjective fractional programming with n-set functions’, Mathematical and Computer Modelling, Vol. 54, pp.490–496. Qi, H. (2009) ‘Local duality of nonlinear semidefinite programming’, Mathematics of Operations Research, Vol. 34, No. 1, pp.124–141. Saini, H. and Gulati, T.R. (2011) ‘Nondifferentiable multiobjective symmetric duality with F-convexity over cones’, Nonlinear Analysis: Theory, Methods & Applications, Vol. 74, pp.1577–1584. Schachinger, W. and Bomze, I. (2009) ‘A conic duality Frank-Wolfe-type theorem via exact penalization in quadratic optimization’, Mathematics of Operations Research, Vol. 34, No. 1, pp.83–91. Sowlati, T., Assadi, P. and Paradi, J.C. (2010) ‘Developing a mathematical programming model for senstivity analysis in analytic hierarchy process’, International Journal of Mathematics in Operational Research, Vol. 2, pp.290–301. Suneja, S.K. and Gupta, S. (1998) ‘Duality in multiobjective nonlinear programming involving semilocally convex and related functions’, European Journal of Operational Research, Vol. 107, pp.675–685. Suneja, S.K., Aggarwal, S. and Davar, S. (2002) ‘Multiobjective symmetric duality involving cones’, European Journal of Operational Research, Vol. 141, pp.471–479. Suneja, S.K., Lalitha, C.S. and Khurana, S. (2003) ‘Second-order symmetric duality in multiobjective programming’, European Journal of Operational Research, Vol. 144, pp.492–500. Weir, T. and Mond, B. (1988) ‘Symmetric and self duality in multiple objective programming’, Asia Pacific Journal of Operational Research, Vol. 5, pp.124–133. Wolfe, P. (1961) ‘A duality theorem for nonlinear programming’, Quarterly of Applied Mathematics, Vol. 19, pp.239–244. Xidonas, P., Mavrotas, G. and Psarras, J. (2010) ‘A multicriteria decision making approach for the evaluation of equity portfolios’, International Journal of Mathematics in Operational Research, Vol. 2, pp.40–72. Yang, X.M. and Hou, S.H. (2001) ‘Second-order symmetric duality in multiobjective programming’, Applied Mathematics Letters, Vol. 14, pp.587–592. Yang, X.M., Yang, X.Q., Teo, K.L. and Hou, S.H. (2005) ‘Second order symmetric duality in non-differentiable multiobjective programming with F-convexity’, European Journal of Operational Research, Vol. 164, pp.406–416. Zheng, X.Y. and Yang, X. (2007) ‘Lagrange multipliers in nonsmooth semi-infinite optimization problems’, Mathematics of Operations Research, Vol. 32, No. 1, pp.168–181.
