Nonlinear Separation Approach to Constrained ... - Semantic Scholar
J Optim Theory Appl (2012) 154:842–856 DOI 10.1007/s10957-012-0027-4
Nonlinear Separation Approach to Constrained Extremum Problems S.J. Li · Y.D. Xu · S.K. Zhu
Received: 14 June 2011 / Accepted: 12 March 2012 / Published online: 24 March 2012 © Springer Science+Business Media, LLC 2012
Abstract In this paper, by virtue of a nonlinear scalarization function, two nonlinear weak separation functions, a nonlinear regular weak separation function, and a nonlinear strong separation function are first introduced, respectively. Then, by the image space analysis, a global saddle-point condition for a nonlinear function is investigated. It is shown that the existence of a saddle point is equivalent to a nonlinear separation of two suitable subsets of the image space. Finally, some necessary and sufficient optimality conditions are obtained for constrained extremum problems. Keywords Image space analysis · Nonlinear (regular) weak separation function · Nonlinear strong separation function · Saddle point
1 Introduction In the Image Space Analysis (ISA), the optimality condition for the constrained extremum problem is expressed under the form of the impossibility of a parametric system, or equivalently, as the disjunction of two suitable subsets K and H of the Communicated by Jafar Zafarani. S.J. Li () · Y.D. Xu · S.K. Zhu College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China e-mail: [email protected] Y.D. Xu e-mail: [email protected] S.K. Zhu e-mail: [email protected] S.J. Li Mathematical Sciences Research Institute in Chongqing, Chongqing University, Chongqing 401331, China
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Image Space (IS) associated with the constrained extremum problem, where K is defined by using the images of the functions involved in the constrained extremum problem, while H is a convex cone that is identified by the kinds of constraints (equalities/inequalities or other conditions). The disjunction between K and H can be proved by showing that they lie in two disjoint level sets of a suitable, possibly nonlinear, separation function. In the constrained extremum problem, several theoretical aspects can be developed, as duality, existence of optimal solutions, optimality conditions (saddle-point sufficient conditions, Lagrangian-type necessary conditions), penalty methods and regularity; see [1–10]. In [2], some optimality conditions have been deduced, e.g., saddle-point conditions and regularity conditions, and some theorems of the alternative. In [4], the relationships between Wolfe and Mond–Weir duality in the IS under suitable generalized convexity assumptions have been analyzed. The ISA can also be extended to classes of problems more general than a constrained extremum problem, as vector optimization problems, vector variational inequalities, and vector equilibrium problems, e.g., see [11–13]. In [13], Mastroeni extended the ISA to a vector quasiequilibrium problem (VQEP) with a variable ordering relation and analyzed scalar and vector saddle point optimality conditions, arising from the existence of a vector separation in the IS associated with VQEP. The nonlinear scalarization function Δ was introduced by Hiriart–Urruty [14], who used it to analyze the geometry of nonsmooth optimization problems and obtained necessary optimality conditions for nonconvex optimization problems. The scalarization method was later extended by Miglierina [15] to deal with set-valued optimization problems and obtain the equivalent relation between the well-posedness of a scalarization problem and a properly efficient solution of a set-valued optimization problem. The bridge role of the nonlinear scalarization function Δ in characterizing the various proper efficiencies of a set was shown in [16]. In this paper, we use the nonlinear scalarization function Δ for constructing two nonlinear weak separation functions, a nonlinear regular weak separation function, and a nonlinear strong separation function in the IS, respectively. Then, by virtue of these nonlinear separation functions and the ISA, we obtain some necessary and sufficient optimality conditions, in particular, a saddle-point sufficient optimality condition for the constrained extremum problem. The paper is organized as follows. In Sect. 2, we recall the main notions and definitions concerning the ISA. In Sect. 3, we introduce two nonlinear weak separation functions, a nonlinear regular weak separation function, and a nonlinear strong separation function, respectively. In Sect. 4, by virtue of these nonlinear separation functions, we obtain a saddle-point sufficient optimality condition and some necessary and sufficient optimality conditions associated with the constrained extremum problem.
2 Preliminaries In this section, we shall recall some notations and definitions, which will be used in the sequel. Let Y be a normed space. The closure, the complement, the relative
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interior, the interior, and the boundary of a set M ⊆ Y are denoted by cl M, M c , ri M, int M, and ∂M, respectively. ·, · denotes the scalar product in Rn , while C ∗ := z ∈ Rn : z, y ≥ 0, ∀y ∈ C , is the positive polar of a cone C ⊆ Rn with apex at the origin. For a function f defined on a set X and α ∈ R, the sets lev≥α f := x ∈ X : f (x) ≥ α , lev>α f := x ∈ X : f (x) > α , are called nonnegative and positive level sets of the function f , respectively. Consider the following constrained extremum problem: f ↓ := min f (x), s.t. x ∈ R := x ∈ X : g(x) ∈ D ,
(1)
where X is a nonempty subset of Rn , f : X → R, g : X → Rm and D is a closed and convex cone in Rm with apex at the origin. We recall the main features of the ISA for the problem (1). Let x¯ ∈ X; define the map Ax¯ : X → R1+m , Ax¯ (x) := f (x) ¯ − f (x), g(x) , and consider the sets Kx¯ := (u, v) ∈ R1+m : (u, v) = Ax¯ (x), x ∈ X , H := (u, v) ∈ R1+m : u > 0, v ∈ D . Kx¯ is called the image of the problem (1), while R1+m is the image space (IS). Obviously, x¯ ∈ R is an optimal solution for the problem (1), if and only if the generalized system Ax¯ (x) ∈ H,
x ∈ X,
(2)
has no solutions, or equivalently, Kx¯ ∩ H = ∅. In the sequel, Π will denote a set of parameters to be specified case by case. Definition 2.1 [3] The class of all the functions w : R1+m × Π → R, such that (i) lev ≥0 w(·; π) ⊇ H, ∀π ∈ Π , (ii) π∈Π lev>0 w(·; π) ⊆ H, is called the class of weak separation functions and is denoted by W (Π). Conditions (i) and (ii) of Definition 2.1 can be strengthened as follows. Definition 2.2 [3] The class of all the functions w : R1+m × Π → R, such that lev>0 w(·; π) = H, π∈Π
is denoted by WR (Π) and is called the class of regular weak separation functions.
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Besides the weak separation functions, another type of separation functions has been introduced in [3]. Definition 2.3 [3] The class of all the functions s : R1+m × Π → R, such that (i) lev >0 s(·; π) ⊆ H, ∀π ∈ Π , (ii) π∈Π lev>0 s(·; π) = ri H, is called the class of strong separation functions and will be denoted by S(Π). Remark 2.1 ri H = int H when int H = ∅. Hence, π∈Π lev>0 s(·; π) = ri H is equivalent to π∈Π lev>0 s(·; π) = int H when int H = ∅. Definition 2.4 [17] Let f (x) be locally Lipschitz on Rn and let υ be any vector in Rn . The generalized directional derivative of f (x) at x ∈ Rn in the direction υ is f (y + tυ) − f (y) . f 0 (x; υ) := lim sup t y→x,t↓0 Definition 2.5 [17] Let f (x) be locally Lipschitz on Rn . The generalized gradient of f (x) at x ∈ Rn is ∂f (x) := ξ ∈ Rn : ξ, d ≤ f 0 (x; d), ∀d ∈ Rn . Lemma 2.1 [17] Let f (x) be locally Lipschitz on Rn ; if x ∗ is a minimum (maximum) point of f (x), then 0Rn ∈ ∂f (x ∗ ). Definition 2.6 [14] Let Y be a normed space. For a set A ⊆ Y , let the function ΔA : Y → R ∪ {±∞} be defined as ΔA (y) := dA (y) − dY \A (y), where dA (y) := infz∈A y − z, d∅ (y) := +∞, and y denotes the norm of y in Y . Geometrically, the function ΔA means that, given a point y ∈ Y , if y ∈ A, then ΔA (y) equals the negative number of the distance between the point y and the set Y \A, i.e., −dY \A (y), and if y ∈ / A, then ΔA (y) equals the distance between the point y and the set A, i.e., dA (y). In general, the function Δ is introduced for analyzing the geometry of nonsmooth optimization problems and obtaining some necessary optimality conditions; see [14, 16, 18, 19]. Its main properties are gathered together in the following proposition. Proposition 2.1 (See Proposition 3.2 in [16]) If the set A is nonempty and A = Y , then (i) (ii) (iii) (iv) (v) (vi)
ΔA is real valued; ΔA is 1-Lipschitzian; ΔA (y) < 0 for every y ∈ int A; ΔA (y) = 0 for every y ∈ ∂A; ΔA (y) > 0 for every y ∈ int Ac ; ΔA is positively homogeneous, if A is a cone with apex at the origin.
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3 Separation Functions In this section, by the ISA, we introduce the following three nonlinear functions and discuss their properties: w1 (u, v; θ, λ) := θ u − ΔR+ λ, v , (u, v) ∈ R1+m , (θ, λ) ∈ Π1 , (3) where Π1 := H∗ \{0Rm+1 }; w2 (z; π) := −ΔH (z + π),
z = (u, v) ∈ R1+m , π ∈ Π2 ,
(4)
where Π2 := cl H; s(z; π) := −ΔH (z − π),
z = (u, v) ∈ R1+m , π ∈ Π2 .
(5)
Proposition 3.1 (i) The nonlinear function w1 is a weak separation function; (ii) The nonlinear function w1 is a regular weak separation function when (θ, λ) ∈ Π3 = ]0, +∞[ ×D ∗ ; (iii) the nonlinear function w2 is a weak separation function; (iv) if int H = ∅, the nonlinear function s is a strong separation function. Proof (i) Set Π = Π1 = H∗ \{0Rm+1 }. Then, for any (θ, λ) ∈ H∗ \{0Rm+1 }, (u, v) ∈ H, we have θ u ≥ 0 and λ, v ≥ 0. By Proposition 2.1(iii), (iv), we have w1 (u, v; θ, λ) = θ u − ΔR+ λ, v ≥ 0, which implies that lev≥0 w1 (·; θ, λ) ⊇ H, We now prove the inclusion
∀(θ, λ) ∈ Π.
lev>0 w1 (·; θ, λ) ⊆ H.
(6)
(7)
(θ,λ)∈Π
Ab absurdo, assume that (7) be false. Then there exists (u, v) ∈ / H, such that w1 (u, v; θ, λ) > 0,
∀(θ, λ) ∈ Π.
(8)
For (u, v) ∈ / H, we consider the following two cases. Case 1. If u ≤ 0 and v ∈ Rm , then, we set θ = 1, λ = 0Rm . So, we have (θ, λ) ∈ Π . But w1 (u, v; θ, λ) = θ u = u ≤ 0, which contradicts (8). Case 2. If u > 0 and v ∈ / D, then there exists λ ∈ D ∗ \{0Rm }, such that λ, v < 0. Set θ = 0, it follows from Proposition 2.1(v) that w1 (u, v; θ, λ) = −ΔR+ λ, v < 0, which contradicts (8).
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So, from (6) and (7), we have w1 ∈ W (Π1 ). (ii) Set Π = Π3 = ]0, +∞[ ×D ∗ . By previous part (i), we have lev>0 w1 (·; π) ⊇ H.
(9)
π∈Π
We now prove the inclusion
lev>0 w1 (·; π) ⊆ H.
(10)
π∈Π
Ab absurdo, assume that (10) be false. Then there exists (u, v) ∈ / H, such that w1 (u, v; θ, λ) > 0,
∀(θ, λ) ∈ Π.
(11)
For (u, v) ∈ / H, we also consider the following two cases. Case 1. If u ≤ 0 and v ∈ Rm , then we set θ = 1, λ = 0Rm . So, we have (θ, λ) ∈ Π . But w1 (u, v; θ, λ) = θ u ≤ 0, which contradicts (11). Case 2. If u > 0 and v ∈ / D, then there exists λ ∈ D ∗ \{0Rm }, such that λ, v < 0. ∗ Since tλ ∈ D \{0Rm }, for any t > 0, by Proposition 2.1(v), (vi), we have w1 (u, v; θ, tλ) = θ u − ΔR+ tλ, v = θ u − tΔR+ λ, v → −∞(t → +∞), which contradicts (11). So, from (9) and (10), we have that w1 ∈ WR (Π3 ). (iii) Set Π = Π2 = cl H. For any (u, v) ∈ H, any π ∈ Π , and H + cl H = H, we have (u, v) + π ∈ H + cl H = H. By Proposition 2.1(iii), (iv), we have w2 (u, v; π) ≥ 0, i.e., lev≥0 w2 (·; π) ⊇ H, Now, we prove the inclusion
∀π ∈ Π.
lev>0 w2 (·; π) ⊆ H.
(12)
π∈Π
In fact, let π0 = 0Rm+1 ∈ cl H. Then, by Proposition 2.1(iii)–(v), we have lev>0 w2 (·; π0 ) = int H ⊆ H. Indeed, by Proposition 2.1(iii), we have lev>0 w2 (·; π0 ) ⊇ int H. Now, we prove lev>0 w2 (·; π0 ) ⊆ int H.
(13)
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If there exists (u, v) ∈ lev>0 w2 (·; π0 ), such that (u, v) ∈ / int H, then, (u, v) ∈ ∂H ∪ int Hc . Hence, by Proposition 2.1(iv), (v), we can obtain (u, v) ∈ lev≤0 w2 (·; π0 ), which contradicts (u, v) ∈ lev>0 w2 (·; π0 ). Therefore, (13) holds. Since lev>0 w2 (·; π) ⊆ lev>0 w2 (·; π0 ), π∈Π
and (13), we have
lev>0 w2 (·; π) ⊆ H.
π∈Π
Hence, we have that w2 ∈ W (Π2 ). (iv) Set Π = Π2 = cl H. Firstly, we prove that lev>0 s(·; π) ⊆ int H,
∀π ∈ Π.
(14)
In fact, for every π ∈ Π , (u, v) ∈ lev>0 s(·; π), by Proposition 2.1(iv), (v), we have (u, v) − π ∈ int H. Indeed, if (u, v) − π ∈ / int H, then (u, v) − π R
m+1
(15)
∈ ∂H ∪ int Hc ,
since
= int H ∪ ∂H ∪ int H . c
But, by Proposition 2.1(iv), if (u, v) − π ∈ ∂H, then s((u, v); π) = 0, which contradicts (u, v) ∈ lev>0 s(·; π). By Proposition 2.1(v), if (u, v) − π ∈ int Hc , then s((u, v); π) < 0, which contradicts (u, v) ∈ lev>0 s(·; π). Therefore, (15) is valid. From (15), we can obtain (u, v) ∈ int H + π ⊆ int H + cl H = int H. So, (14) is true. Secondly, we show that
lev>0 s(·; π) ⊆ int H.
(16)
π∈Π
In fact, let π0 = 0Rm+1 ∈ cl H. Then, by Proposition 2.1(iii)–(v), we have lev>0 s(·; π0 ) = int H. So, int H = lev>0 s(·; π0 ) ⊆
lev>0 s(·; π).
π∈Π
By (14), we have
lev>0 s(·; π) = int H.
π∈Π
It follows from (14) and (16) that the nonlinear function s is a strong separation function, i.e., the nonlinear function s ∈ S(Π2 ).
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4 Optimality Conditions In this section, we establish some weak and strong theorems of the alternative and then discuss the optimality conditions for the problem (1). Theorem 4.1 (i) The system (2) in the unknown x, and the system:
∃(θ, λ) ∈ Π1 s.t. θ f (x) ¯ − f (x) − ΔR+ λ, g(x) < 0,
∀x ∈ X, (17)
are not simultaneously possible. (ii) The system (2) in the unknown x, and the system: ∃π ∈ Π2 s.t. ΔH f (x) ¯ − f (x), g(x) + π > 0,
∀x ∈ X,
(18)
are not simultaneously possible. (iii) The system (2) in the unknown x, and the system: ∃π ∈ Π2 s.t. ΔH f (x) ¯ − f (x), g(x) − π ≥ 0,
∀x ∈ X,
(19)
are not simultaneously impossible. Proof From Proposition 3.1, weak and strong theorems of the alternative in [3], we know that the above results hold. Theorem 4.2 The system (2) in the unknown x, and the system:
∃(θ, λ) ∈ ]0, +∞[ ×D ∗ s.t. θ f (x) ¯ − f (x) − ΔR+ λ, g(x) ≤ 0,
∀x ∈ X, (20)
are not simultaneously possible. Proof The proof is similar to the one of Theorem 4.4.2 in [3], so we omit it here. By Theorems 4.1 and 4.2, we can obtain some sufficient and necessary optimality conditions for the problem (1). ¯ ∈ H∗ \{0Rm+1 }, such that Theorem 4.3 (i) Let x¯ ∈ R be given. If there exists (θ¯ , λ)
¯ g(x) < 0, ∀x ∈ X, (21) θ¯ f (x) ¯ − f (x) − ΔR+ λ, then x¯ is a global solution of the problem (1). ¯ ∈ H∗ with θ¯ > 0, such that (ii) Let x¯ ∈ R be given. If there exists (θ¯ , λ)
¯ g(x) ≤ 0, ∀x ∈ X, θ¯ f (x) ¯ − f (x) − ΔR+ λ,
(22)
then x¯ is a global solution of the problem (1). (iii) Let x¯ ∈ R be given. If there exists π ∈ cl H, such that ΔH f (x) ¯ − f (x), g(x) + π > 0, ∀x ∈ X,
(23)
then x¯ is a global solution of the problem (1). (iv) If x¯ is a global solution of the problem (1) and int H = ∅, then there exists π ∈ cl H, such that ΔH f (x) ¯ − f (x), g(x) − π ≥ 0, ∀x ∈ X. (24)
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Proof (i) (21) ⇒ the system (17) is possible. By Theorem 4.1(i), we know that the system (2) is impossible, i.e., Kx¯ ∩ H = ∅. Since x¯ ∈ R, x¯ is a global solution of the problem (1). (ii) (22) ⇒ the system (20) is possible. By Theorem 4.2, we know that the system (2) is impossible, i.e., Kx¯ ∩ H = ∅. Since x¯ ∈ R, x¯ is a global solution of the problem (1). (iii) (23) ⇒ the system (18) is possible. By Theorem 4.1(ii), we know that the system (2) is impossible, i.e., Kx¯ ∩ H = ∅. Since x¯ ∈ R, x¯ is a global solution of the problem (1). (iv) Since x¯ is a global solution of the problem (1), the system (2) is impossible. By Theorem 4.1(iii), we have that the system (19) is possible, namely, there exists π ∈ cl H, such that s Ax¯ (x); π = −ΔH f (x) ¯ − f (x), g(x) − π ≤ 0, ∀x ∈ X,
that is, (24) holds.
Now, we give Examples 4.1 and 4.2 to illustrate Theorem 4.3(ii) and (iv), respectively. Example 4.1 In the problem (1), set X := {(x1 , x2 ) ∈ R2 : x12 + x22 ≤ 1}, D = R+ , f (x) = x1 + 2x2 + 2, g(x) = 1√+ x1 +√ x2 . √ ¯ − f (x) = − 5 − x1 − 2x2 . Consider the vector x¯ = (− 55 , − 2 5 5 ) ∈ R, then f (x) Set θ¯ = 1 and λ¯ = 1, so we have √ w1 Ax¯ (x); θ¯ , λ¯ = − 5 − x1 − 2x2 − ΔR+ (1 + x1 + x2 ) √ = − 5 − x1 − 2x2 + x1 + x2 + 1 √ = − 5 + 1 − x2 ≤ 0, ∀(x1 , x2 ) ∈ X, √
√
that is, (22) is valid. We can compute that x¯ = (− 55 , − 2 5 5 ) is a global solution of the problem (1) by simplex method for linear programming. Example√4.2 In√this example, we shall continue to consider Example 4.1. We see that x¯ = (− 55 , − 2 5 5 ) is a global solution of the problem (1). There exists π = (2, 0) ∈ cl H, such that √ f (x) ¯ − f (x), g(x) − π = (− 5 − 2 − x1 − 2x2 , x1 + x2 + 1) ∈ / H, √ since − 5−2−x1 −2x2 < 0 for all (x1 , x2 ) ∈ X. By Proposition 2.1(iv), (v), we have ΔH f (x) ¯ − f (x), g(x) − π ≥ 0, ∀x ∈ X, which implies that (24) be valid. Using the nonlinear regular weak separation function in (3), we can get the following sufficient and necessary optimality condition for the problem (1). Theorem 4.4 Let θ¯ ∈ ]0, +∞[ be given. Let ¯ λ) := θ¯ u − ΔR+ λ, v , w3 (u, v; θ,
(u, v) ∈ R1+m , λ ∈ D ∗ .
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Then x¯ ∈ R is a global solution of the problem (1) if and only if
sup inf ∗ θ¯ f (x) ¯ − f (x) − ΔR+ λ, g(x) = 0.
(25)
Proof First, we prove ¯ λ) ⊇ lev>0 inf w3 (·; θ, ¯ λ) ⊇ H. H= lev>0 w3 (·; θ, ∗
(26)
x∈X λ∈D
λ∈D
λ∈D ∗
In fact, the first equality of (26) holds obviously, since function w3 is a regular weak separation function. The second inclusion of (26) is also true obviously. Now, we show that ¯ λ) ⊇ H. lev>0 inf ∗ w3 (·; θ, λ∈D
In fact, for all (u, v) ∈ H,
λ ∈ D∗ ,
by Proposition 2.1(iii), (iv), we have ΔR+ λ, v ≤ 0.
¯ λ) = θ¯ u > 0, i.e., Then, infλ∈D ∗ w3 (u, v; θ, ¯ λ) ⊇ H. lev>0 inf ∗ w3 (·; θ, λ∈D
From (26), we have ¯ λ). H = lev>0 inf ∗ w3 (·; θ,
(27)
λ∈D
If x¯ ∈ R is a global solution of the problem (1), then H ∩ Kx¯ = ∅. It follows from the equality (27) that ¯ λ) ∩ Kx¯ = ∅, lev>0 inf ∗ w3 (·; θ, λ∈D
which implies that
¯ − f (x) − ΔR+ λ, g(x) ≤ 0, inf θ¯ f (x) λ∈D ∗
Moreover,
∀x ∈ X.
inf ∗ θ¯ × 0 − ΔR+ λ, g(x) ¯ ≥ 0,
λ∈D
because
¯ λ). 0, g(x) ¯ ∈ cl H ⊆ lev≥0 inf ∗ w3 (·; θ, λ∈D
Therefore, one gets
¯ = 0, inf∗ θ¯ × 0 − ΔR+ λ, g(x)
λ∈D
which together with inequality (28), gives the equality (25). Conversely, suppose that equality (25) holds. Then one has
¯ − f (x) − ΔR+ λ, g(x) ≤ 0, ∀x ∈ X. inf∗ θ¯ f (x) λ∈D
(28)
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Simultaneously, we have ¯ λ) ⊇ H. lev>0 inf ∗ w3 (·; θ, λ∈D
So, the relation Kx¯ ∩ H = ∅ holds, which allows us to conclude that x¯ ∈ R is a global solution of the problem (1). The following example is given to illustrate Theorem 4.4. Example√ 4.3 In√ this example, we continue to consider Example 4.1. We know that x¯ = (− 55 , − 2 5 5 ) is a global solution of the problem (1) in Example 4.1. Let θ¯ ∈ ]0, +∞[ be given. By using direct computation, we can get ⎧ if x = x, ¯
⎨ 0, inf∗ θ¯ f (x) ¯ − f (x) − ΔR+ λ, g(x) = θ¯ (f (x) ¯ − f (x)), if x ∈ R\{x}, ¯ ⎩ λ∈D −∞, otherwise. So, supx∈X infλ∈D ∗ [θ¯ (f (x) ¯ − f (x)) − ΔR+ (λ, g(x))] = 0. On the other hand, if supx∈X infλ∈D ∗ [θ¯ (f (x) ¯ − f (x)) − ΔR+ (λ, g(x))] = 0, then
inf ∗ θ¯ f (x) ¯ − f (x) − ΔR+ λ, g(x) ≤ 0, ∀x ∈ X. λ∈D
In particular, let x ∈ R; we have
inf ∗ θ¯ f (x) ¯ − f (x) ≤ 0, ¯ − f (x) − ΔR+ λ, g(x) = θ¯ f (x) λ∈D
which implies that f (x) ¯ ≤ f (x),
∀x ∈ R.
The impossibility of the system (2) can be stated by means of separation arguments in the IS, viz, the sets Kx¯ and H of the IS lie in disjoint level sets of a suitable nonlinear function. Now, we introduce a concept related to the sets Kx¯ and H, and then obtain some optimality conditions of the problem (1). Definition 4.1 The sets Kx¯ and H admit a nonlinear separation, if and only if there ¯ ∈ H∗ \{0Rm+1 }, such that exists (θ¯ , λ)
¯ g(x) ≤ 0, x ∈ X. w1 Ax¯ (x); θ¯ , λ¯ = θ¯ f (x) ¯ − f (x) − ΔR+ λ, (29) Moreover, the separation is said to be regular iff θ¯ > 0 in (29). Proposition 4.1 If Kx¯ and H admit a regular nonlinear separation, then x¯ ∈ R is a global solution of the problem (1). Proof Since
(θ,λ)∈ ]0,+∞[×D ∗
lev>0 w1 (·; θ, λ) ⊇ H and (29) holds, we have Kx¯ ∩ H = ∅.
Hence, we can conclude that x¯ ∈ R is a global solution of the problem (1).
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In order to obtain a saddle-point sufficient optimality condition associated with the problem (1), we consider the following nonlinear function
L(x; θ, λ) := θf (x) + ΔR+ λ, g(x) , x ∈ X, (θ, λ) ∈ H∗ \{0Rm+1 }. (30) Theorem 4.5 The following statements are equivalent: (i) x¯ ∈ R, and Kx¯ and H admit a nonlinear separation; ¯ is a saddle point for L(x; θ, ¯ λ) ¯ ∈ H∗ \{0Rm+1 }, such that (x, ¯ λ) (ii) There exists (θ¯ , λ) ∗ on X × D , i.e., ¯ λ) ≤ L(x; L(x; ¯ θ, ¯ θ¯ , λ¯ ) ≤ L(x; θ¯ , λ¯ ),
∀x ∈ X, ∀λ ∈ D ∗ .
(31)
Proof (i) ⇒ (ii). Suppose that Kx¯ and H admit a nonlinear separation. Then there ¯ ∈ H∗ \{0Rm+1 } such that (29) holds. From (29), we can obtain that exists (θ¯ , λ)
¯ g(x) ΔR+ λ, ¯ ≥ 0. ¯ g(x)) ¯ = 0, since g(x) ¯ ∈ D. ThereBy Proposition 2.1(iii), (iv), we can see ΔR+ (λ, fore,
¯ g(x) ¯ λ) = θ¯ f (x) ¯ ≤ θ¯ f (x) ¯ + ΔR+ λ, ¯ L(x; ¯ θ, ¯ + ΔR+ λ, g(x) ¯ = L(x; ¯ θ¯ , λ), ∀λ ∈ D ∗ , since ΔR+ (λ, g(x)) ¯ ≤ 0 for all λ ∈ D ∗ . Now, we prove the second inequality of (31). In fact, since (29) holds, we have
¯ ∀x ∈ X. θ¯ f (x) + ΔR+ λ¯ , g(x) ≥ θ¯ f (x), ¯ = 0 that It follows from ΔR+ (λ¯ , g(x))
¯ g(x) ≥ θ¯ f (x) ¯ g(x) θ¯ f (x) + ΔR+ λ, ¯ + ΔR+ λ, ¯ ,
∀x ∈ X,
which is the second inequality of (31). ¯ be a saddle point for L(x; θ, ¯ λ) on X × D ∗ , that is, (ii) ⇒ (i). Suppose that (x, ¯ λ)
θ¯ f (x) ¯ + ΔR+ λ, g(x) ¯ ≤ θ¯ f (x) ¯ + ΔR+ λ¯ , g(x) ¯
≤ θ¯ f (x) + ΔR+ λ¯ , g(x) , ∀x ∈ X, ∀λ ∈ D ∗ . First of all, we prove that x¯ ∈ R. Ab absurdo, we suppose that g(x) ¯ ∈ / D. Then there ¯ < 0. Since D ∗ is a cone, we have exists λ∗ ∈ D ∗ , such that λ∗ , g(x)
tλ∗ ∈ D ∗ , ∀t ≥ 0 and t λ∗ , g(x) ¯ → −∞, t → +∞. By Proposition 2.1(v), (vi), we get
tΔR+ λ∗ , g(x) ¯ → +∞,
t → +∞,
which contradicts the first inequality. Taking λ = 0, it follows from the first inequality ¯ g(x)) ¯ g(x)) ¯ ≥ 0. Since x¯ ∈ R, ΔR+ (λ, ¯ ≤ 0 by Proposition 2.1(iii), that ΔR+ (λ, (iv). Therefore,
¯ g(x) ΔR+ λ, ¯ = 0. The second inequality coincides with (29) and the proof is complete.
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Remark 4.1 In the definition of the function L(x; θ, λ), we only need (θ, λ) ∈ H∗ \{0Rm+1 }. If we further assume θ > 0, then, without any loss of generality, we can set θ = 1. Since the function −ΔR+ (λ, g(x)) satisfies the properties of the function w(g(x); γ ) in (5.2.10.a) of [3], the nonlinear function L fulfills the hypotheses of Proposition 5.2.11 of [3]. So, in this case, Theorem 4.5 is a special case of Proposition 5.2.11. However, in the definition of the function L(x; θ, λ), θ may take 0. So, Theorem 4.5, in fact, is different from Proposition 5.2.11 of [3]. ¯ is a Corollary 4.1 Let x¯ ∈ X. If there exists θ¯ ∈ ]0, +∞[ and λ¯ ∈ D ∗ , such that (x, ¯ λ) ∗ ¯ saddle point for L(x; θ , λ) on X × D , then x¯ is a global solution of the problem (1). Proof From Theorem 4.5 and θ¯ ∈ ]0, +∞[, we have that the sets Kx¯ and H admit a regular nonlinear separation, and x¯ ∈ R. By Proposition 4.1, we obtain that x¯ is a global solution of the problem (1). Remark 4.2 The condition that θ¯ ∈ ]0, +∞[ in Corollary 4.1 is necessary, i.e., if there ¯ is a saddle point for L(x; θ¯ , λ) on ¯ λ) exists θ¯ = 0 and λ¯ ∈ D ∗ \{0Rm } such that (x, ∗ X × D , then x¯ ∈ X may be not a global solution of the problem (1). The following example is given to illustrate this case. Example 4.4 In the problem (1), set X = R, D = R2+ , f (x) = −3 + x, g1 (x) = −1 + x 2 , g2 (x) = x 2 (1 − x 2 ). ¯ with x¯ = 1, and λ¯ = (1, 1) is a saddle of L(x; 0, λ). Then we can get (x, ¯ λ) But x¯ = 1 is not a global solution of the problem (1). Hence, the condition that θ¯ ∈ ]0, +∞[ in Corollary 4.1 is necessary. ¯ ∈ H∗ \{0Rm+1 }. Assume that the Theorem 4.6 Let X be an open set in Rn and (θ¯ , λ) real valued function f (·) be locally Lipschitz on X. Suppose that the vector valued function g(·) be locally Lipschitz on X, i.e., for any x ∈ X, there exist a constant L and a neighborhood Nx of x, such that g(x1 ) − g(x2 ) ≤ Lx1 − x2 , ∀x1 , x2 ∈ Nx . ¯ is a saddle point for L(x; θ¯ , λ) on X × D ∗ , then If (x, ¯ λ) ⎧ ¯ λ), ¯ 0Rn ∈ ∂Lλ (x; ¯ θ, ⎪ ⎪ ⎪ ⎪ ⎨ 0 m ∈ ∂L (x; ¯ λ), ¯ x ¯ θ, R ¯ g(x)) ⎪ ¯ = 0, ΔR+ (λ, ⎪ ⎪ ⎪ ⎩ x¯ ∈ R. ¯ λ) on X × D ∗ , that is, Proof Suppose that (x, ¯ λ¯ ) be a saddle point for L(x; θ,
θ¯ f (x) ¯ + ΔR+ λ, g(x) ¯ ≤ θ¯ f (x) ¯ + ΔR+ λ¯ , g(x) ¯
≤ θ¯ f (x) + ΔR+ λ¯ , g(x) , ∀x ∈ X, ∀λ ∈ D ∗ . As in the proof of the sufficiency in Theorem 4.5, we can show that
¯ g(x) ¯ = 0. x¯ ∈ R and ΔR+ λ,
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¯ λ). We The first inequality implies that λ¯ be a global maximum solution of L(x; ¯ θ, now claim that the nonlinear function L(x; ¯ θ¯ , λ) is Lipschitz on D ∗ . In fact, for any λ1 , λ2 ∈ D ∗ , by Proposition 2.1(ii), we have
L(x; ¯ λ1 ) − L(x; ¯ λ2 ) = ΔR+ λ1 , g(x) ¯ θ, ¯ θ, ¯ − ΔR+ λ2 , g(x) ¯
≤ λ1 − λ2 , g(x) ¯ ¯ λ1 − λ2 . ≤ g(x) Hence, the nonlinear function L(x; ¯ θ¯ , λ) is Lipschitz on D ∗ . Therefore, by Lemma 2.1, one has 0Rm ∈ ∂Lλ (x; ¯ θ¯ , λ¯ ). ¯ We The second inequality implies that x¯ be a global minimum solution of L(x; θ¯ , λ). ¯ ¯ now claim that the nonlinear function L(x; θ, λ) is locally Lipschitz on X. Indeed, for any x ∈ X, since the functions f (·) and g(·) are locally Lipschitz on X, then there exists a neighborhood Nx of x, such that L(x1 ; θ, ¯ λ) ¯ − L(x2 ; θ¯ , λ) ¯
¯ g(x1 ) − ΔR+ λ, ¯ g(x2 ) = θ¯ f (x1 ) − θ¯ f (x2 ) + ΔR+ λ,
¯ g(x1 ) − ΔR+ λ, ¯ g(x2 ) ≤ θ¯ f (x1 ) − f (x2 ) + ΔR+ λ,
¯ g(x1 ) − g(x2 ) ≤ θ¯ L1 x1 − x2 + λ, ¯ ≤ θ¯ L1 x1 − x2 + L2 λx 1 − x2 = (θ¯ L1 + λ¯ L2 )x1 − x2 ,
∀x1 , x2 ∈ Nx ,
where L1 , L2 are Lipschitz constants for functions f (·) and g(·) at x, respectively. Hence, it follows from Lemma 2.1 that ¯ λ¯ ). 0Rn ∈ ∂Lx (x; ¯ θ, Thus, the proof is complete.
5 Concluding Remarks We have considered two nonlinear weak separation functions, a nonlinear regular weak separation function, and a nonlinear strong separation function by virtue of nonlinear scalarization function Δ, respectively. We have shown that the existence of a global saddle-point for a nonlinear function is equivalent to a nonlinear separation of two suitable subsets in the IS. Finally, we have proved some necessary and sufficient optimality conditions, in particular, a saddle-point sufficient optimality condition for the problem (1). Acknowledgements This research was supported by the National Natural Science Foundation of China (Grant Numbers: 10871216 and 11171362). The authors thank the two anonymous reviewers for their valuable comments and suggestions, which helped to improve the paper.
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References 1. Dien, P.H., Mastroeni, G., Pappalardo, M., Quang, P.H.: Regularity conditions for constrained extremum problems via image space. J. Optim. Theory Appl. 80, 19–37 (1994) 2. Giannessi, F.: Theorems of the alternative and optimality conditions. J. Optim. Theory Appl. 42, 331– 365 (1984) 3. Giannessi, F.: Constrained Optimization and Image Space Analysis. Springer, Berlin (2005) 4. Giannessi, F., Mastroeni, G.: Separation of sets and Wolfe duality. J. Glob. Optim. 42, 401–412 (2008) 5. Mastroeni, G., Panicucci, B., Passacantando, M., Yao, J.C.: A separation approach to vector quasiequilibrium problems: saddle point and gap function. Taiwan. J. Math. 13, 657–673 (2009) 6. Moldovan, A., Pellegrini, L.: On regularity for constrained extremum problems. Part 1: suffficient optimality conditions. J. Optim. Theory Appl. 142, 147–163 (2009) 7. Moldovan, A., Pellegrini, L.: On regularity for constrained extremum problems. Part 2: necessary optimality conditions. J. Optim. Theory Appl. 142, 165–183 (2009) 8. Pappalardo, M.: Image space approach to penalty methods. J. Optim. Theory Appl. 64, 141–152 (1990) 9. Rubinov, A.M., Uderzo, A.: On global optimality conditions via separation functions. J. Optim. Theory Appl. 109, 345–370 (2001) 10. Tardella, F.: On the image of a constrained extremum problem and some applications to existence of a minimum. J. Optim. Theory Appl. 69, 93–104 (1989) 11. Chinaie, M., Zafarani, J.: Image space analysis and scalarization of multivalued optimization. J. Optim. Theory Appl. 142, 451–467 (2009) 12. Mastroeni, G.: Some applications of the image space analysis to the duality theory for constrained extremum problems. J. Glob. Optim. 46, 603–614 (2010) 13. Mastroeni, G.: On the image space analysis for vector quasi-equilibrium problems with a variable ordering relation. J. Glob. Optim. (2011). doi:10.1007/s10898-011-9674-5 14. Hiriart-Urruty, J.B.: Tangent cone, generalized gradients and mathematical programming in Bananch spaces. Math. Oper. Res. 4, 79–97 (1979) 15. Miglierina, E.: Characterization of solutions of multiobjective optimization problems. Rend. Circ. Mat. Palermo 50, 153–164 (2001) 16. Zaffaroni, A.: Degrees of efficiency and degrees of minimality. SIAM J. Control Optim. 42, 1071– 1086 (2003) 17. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983) 18. Miglierina, E., Molho, E.: Scalarization and stability in vector optimization. J. Optim. Theory Appl. 114, 657–670 (2002) 19. Amahroq, T., Taa, A.: On Lagrange Kuhn–Tucker multipliers for multiobjective optimization problems. Optimization 41, 159–172 (1997)
Nonlinear Separation Approach to Constrained Extremum Problems S.J. Li · Y.D. Xu · S.K. Zhu
Received: 14 June 2011 / Accepted: 12 March 2012 / Published online: 24 March 2012 © Springer Science+Business Media, LLC 2012
Abstract In this paper, by virtue of a nonlinear scalarization function, two nonlinear weak separation functions, a nonlinear regular weak separation function, and a nonlinear strong separation function are first introduced, respectively. Then, by the image space analysis, a global saddle-point condition for a nonlinear function is investigated. It is shown that the existence of a saddle point is equivalent to a nonlinear separation of two suitable subsets of the image space. Finally, some necessary and sufficient optimality conditions are obtained for constrained extremum problems. Keywords Image space analysis · Nonlinear (regular) weak separation function · Nonlinear strong separation function · Saddle point
1 Introduction In the Image Space Analysis (ISA), the optimality condition for the constrained extremum problem is expressed under the form of the impossibility of a parametric system, or equivalently, as the disjunction of two suitable subsets K and H of the Communicated by Jafar Zafarani. S.J. Li () · Y.D. Xu · S.K. Zhu College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China e-mail: [email protected] Y.D. Xu e-mail: [email protected] S.K. Zhu e-mail: [email protected] S.J. Li Mathematical Sciences Research Institute in Chongqing, Chongqing University, Chongqing 401331, China
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Image Space (IS) associated with the constrained extremum problem, where K is defined by using the images of the functions involved in the constrained extremum problem, while H is a convex cone that is identified by the kinds of constraints (equalities/inequalities or other conditions). The disjunction between K and H can be proved by showing that they lie in two disjoint level sets of a suitable, possibly nonlinear, separation function. In the constrained extremum problem, several theoretical aspects can be developed, as duality, existence of optimal solutions, optimality conditions (saddle-point sufficient conditions, Lagrangian-type necessary conditions), penalty methods and regularity; see [1–10]. In [2], some optimality conditions have been deduced, e.g., saddle-point conditions and regularity conditions, and some theorems of the alternative. In [4], the relationships between Wolfe and Mond–Weir duality in the IS under suitable generalized convexity assumptions have been analyzed. The ISA can also be extended to classes of problems more general than a constrained extremum problem, as vector optimization problems, vector variational inequalities, and vector equilibrium problems, e.g., see [11–13]. In [13], Mastroeni extended the ISA to a vector quasiequilibrium problem (VQEP) with a variable ordering relation and analyzed scalar and vector saddle point optimality conditions, arising from the existence of a vector separation in the IS associated with VQEP. The nonlinear scalarization function Δ was introduced by Hiriart–Urruty [14], who used it to analyze the geometry of nonsmooth optimization problems and obtained necessary optimality conditions for nonconvex optimization problems. The scalarization method was later extended by Miglierina [15] to deal with set-valued optimization problems and obtain the equivalent relation between the well-posedness of a scalarization problem and a properly efficient solution of a set-valued optimization problem. The bridge role of the nonlinear scalarization function Δ in characterizing the various proper efficiencies of a set was shown in [16]. In this paper, we use the nonlinear scalarization function Δ for constructing two nonlinear weak separation functions, a nonlinear regular weak separation function, and a nonlinear strong separation function in the IS, respectively. Then, by virtue of these nonlinear separation functions and the ISA, we obtain some necessary and sufficient optimality conditions, in particular, a saddle-point sufficient optimality condition for the constrained extremum problem. The paper is organized as follows. In Sect. 2, we recall the main notions and definitions concerning the ISA. In Sect. 3, we introduce two nonlinear weak separation functions, a nonlinear regular weak separation function, and a nonlinear strong separation function, respectively. In Sect. 4, by virtue of these nonlinear separation functions, we obtain a saddle-point sufficient optimality condition and some necessary and sufficient optimality conditions associated with the constrained extremum problem.
2 Preliminaries In this section, we shall recall some notations and definitions, which will be used in the sequel. Let Y be a normed space. The closure, the complement, the relative
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interior, the interior, and the boundary of a set M ⊆ Y are denoted by cl M, M c , ri M, int M, and ∂M, respectively. ·, · denotes the scalar product in Rn , while C ∗ := z ∈ Rn : z, y ≥ 0, ∀y ∈ C , is the positive polar of a cone C ⊆ Rn with apex at the origin. For a function f defined on a set X and α ∈ R, the sets lev≥α f := x ∈ X : f (x) ≥ α , lev>α f := x ∈ X : f (x) > α , are called nonnegative and positive level sets of the function f , respectively. Consider the following constrained extremum problem: f ↓ := min f (x), s.t. x ∈ R := x ∈ X : g(x) ∈ D ,
(1)
where X is a nonempty subset of Rn , f : X → R, g : X → Rm and D is a closed and convex cone in Rm with apex at the origin. We recall the main features of the ISA for the problem (1). Let x¯ ∈ X; define the map Ax¯ : X → R1+m , Ax¯ (x) := f (x) ¯ − f (x), g(x) , and consider the sets Kx¯ := (u, v) ∈ R1+m : (u, v) = Ax¯ (x), x ∈ X , H := (u, v) ∈ R1+m : u > 0, v ∈ D . Kx¯ is called the image of the problem (1), while R1+m is the image space (IS). Obviously, x¯ ∈ R is an optimal solution for the problem (1), if and only if the generalized system Ax¯ (x) ∈ H,
x ∈ X,
(2)
has no solutions, or equivalently, Kx¯ ∩ H = ∅. In the sequel, Π will denote a set of parameters to be specified case by case. Definition 2.1 [3] The class of all the functions w : R1+m × Π → R, such that (i) lev ≥0 w(·; π) ⊇ H, ∀π ∈ Π , (ii) π∈Π lev>0 w(·; π) ⊆ H, is called the class of weak separation functions and is denoted by W (Π). Conditions (i) and (ii) of Definition 2.1 can be strengthened as follows. Definition 2.2 [3] The class of all the functions w : R1+m × Π → R, such that lev>0 w(·; π) = H, π∈Π
is denoted by WR (Π) and is called the class of regular weak separation functions.
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Besides the weak separation functions, another type of separation functions has been introduced in [3]. Definition 2.3 [3] The class of all the functions s : R1+m × Π → R, such that (i) lev >0 s(·; π) ⊆ H, ∀π ∈ Π , (ii) π∈Π lev>0 s(·; π) = ri H, is called the class of strong separation functions and will be denoted by S(Π). Remark 2.1 ri H = int H when int H = ∅. Hence, π∈Π lev>0 s(·; π) = ri H is equivalent to π∈Π lev>0 s(·; π) = int H when int H = ∅. Definition 2.4 [17] Let f (x) be locally Lipschitz on Rn and let υ be any vector in Rn . The generalized directional derivative of f (x) at x ∈ Rn in the direction υ is f (y + tυ) − f (y) . f 0 (x; υ) := lim sup t y→x,t↓0 Definition 2.5 [17] Let f (x) be locally Lipschitz on Rn . The generalized gradient of f (x) at x ∈ Rn is ∂f (x) := ξ ∈ Rn : ξ, d ≤ f 0 (x; d), ∀d ∈ Rn . Lemma 2.1 [17] Let f (x) be locally Lipschitz on Rn ; if x ∗ is a minimum (maximum) point of f (x), then 0Rn ∈ ∂f (x ∗ ). Definition 2.6 [14] Let Y be a normed space. For a set A ⊆ Y , let the function ΔA : Y → R ∪ {±∞} be defined as ΔA (y) := dA (y) − dY \A (y), where dA (y) := infz∈A y − z, d∅ (y) := +∞, and y denotes the norm of y in Y . Geometrically, the function ΔA means that, given a point y ∈ Y , if y ∈ A, then ΔA (y) equals the negative number of the distance between the point y and the set Y \A, i.e., −dY \A (y), and if y ∈ / A, then ΔA (y) equals the distance between the point y and the set A, i.e., dA (y). In general, the function Δ is introduced for analyzing the geometry of nonsmooth optimization problems and obtaining some necessary optimality conditions; see [14, 16, 18, 19]. Its main properties are gathered together in the following proposition. Proposition 2.1 (See Proposition 3.2 in [16]) If the set A is nonempty and A = Y , then (i) (ii) (iii) (iv) (v) (vi)
ΔA is real valued; ΔA is 1-Lipschitzian; ΔA (y) < 0 for every y ∈ int A; ΔA (y) = 0 for every y ∈ ∂A; ΔA (y) > 0 for every y ∈ int Ac ; ΔA is positively homogeneous, if A is a cone with apex at the origin.
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3 Separation Functions In this section, by the ISA, we introduce the following three nonlinear functions and discuss their properties: w1 (u, v; θ, λ) := θ u − ΔR+ λ, v , (u, v) ∈ R1+m , (θ, λ) ∈ Π1 , (3) where Π1 := H∗ \{0Rm+1 }; w2 (z; π) := −ΔH (z + π),
z = (u, v) ∈ R1+m , π ∈ Π2 ,
(4)
where Π2 := cl H; s(z; π) := −ΔH (z − π),
z = (u, v) ∈ R1+m , π ∈ Π2 .
(5)
Proposition 3.1 (i) The nonlinear function w1 is a weak separation function; (ii) The nonlinear function w1 is a regular weak separation function when (θ, λ) ∈ Π3 = ]0, +∞[ ×D ∗ ; (iii) the nonlinear function w2 is a weak separation function; (iv) if int H = ∅, the nonlinear function s is a strong separation function. Proof (i) Set Π = Π1 = H∗ \{0Rm+1 }. Then, for any (θ, λ) ∈ H∗ \{0Rm+1 }, (u, v) ∈ H, we have θ u ≥ 0 and λ, v ≥ 0. By Proposition 2.1(iii), (iv), we have w1 (u, v; θ, λ) = θ u − ΔR+ λ, v ≥ 0, which implies that lev≥0 w1 (·; θ, λ) ⊇ H, We now prove the inclusion
∀(θ, λ) ∈ Π.
lev>0 w1 (·; θ, λ) ⊆ H.
(6)
(7)
(θ,λ)∈Π
Ab absurdo, assume that (7) be false. Then there exists (u, v) ∈ / H, such that w1 (u, v; θ, λ) > 0,
∀(θ, λ) ∈ Π.
(8)
For (u, v) ∈ / H, we consider the following two cases. Case 1. If u ≤ 0 and v ∈ Rm , then, we set θ = 1, λ = 0Rm . So, we have (θ, λ) ∈ Π . But w1 (u, v; θ, λ) = θ u = u ≤ 0, which contradicts (8). Case 2. If u > 0 and v ∈ / D, then there exists λ ∈ D ∗ \{0Rm }, such that λ, v < 0. Set θ = 0, it follows from Proposition 2.1(v) that w1 (u, v; θ, λ) = −ΔR+ λ, v < 0, which contradicts (8).
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So, from (6) and (7), we have w1 ∈ W (Π1 ). (ii) Set Π = Π3 = ]0, +∞[ ×D ∗ . By previous part (i), we have lev>0 w1 (·; π) ⊇ H.
(9)
π∈Π
We now prove the inclusion
lev>0 w1 (·; π) ⊆ H.
(10)
π∈Π
Ab absurdo, assume that (10) be false. Then there exists (u, v) ∈ / H, such that w1 (u, v; θ, λ) > 0,
∀(θ, λ) ∈ Π.
(11)
For (u, v) ∈ / H, we also consider the following two cases. Case 1. If u ≤ 0 and v ∈ Rm , then we set θ = 1, λ = 0Rm . So, we have (θ, λ) ∈ Π . But w1 (u, v; θ, λ) = θ u ≤ 0, which contradicts (11). Case 2. If u > 0 and v ∈ / D, then there exists λ ∈ D ∗ \{0Rm }, such that λ, v < 0. ∗ Since tλ ∈ D \{0Rm }, for any t > 0, by Proposition 2.1(v), (vi), we have w1 (u, v; θ, tλ) = θ u − ΔR+ tλ, v = θ u − tΔR+ λ, v → −∞(t → +∞), which contradicts (11). So, from (9) and (10), we have that w1 ∈ WR (Π3 ). (iii) Set Π = Π2 = cl H. For any (u, v) ∈ H, any π ∈ Π , and H + cl H = H, we have (u, v) + π ∈ H + cl H = H. By Proposition 2.1(iii), (iv), we have w2 (u, v; π) ≥ 0, i.e., lev≥0 w2 (·; π) ⊇ H, Now, we prove the inclusion
∀π ∈ Π.
lev>0 w2 (·; π) ⊆ H.
(12)
π∈Π
In fact, let π0 = 0Rm+1 ∈ cl H. Then, by Proposition 2.1(iii)–(v), we have lev>0 w2 (·; π0 ) = int H ⊆ H. Indeed, by Proposition 2.1(iii), we have lev>0 w2 (·; π0 ) ⊇ int H. Now, we prove lev>0 w2 (·; π0 ) ⊆ int H.
(13)
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If there exists (u, v) ∈ lev>0 w2 (·; π0 ), such that (u, v) ∈ / int H, then, (u, v) ∈ ∂H ∪ int Hc . Hence, by Proposition 2.1(iv), (v), we can obtain (u, v) ∈ lev≤0 w2 (·; π0 ), which contradicts (u, v) ∈ lev>0 w2 (·; π0 ). Therefore, (13) holds. Since lev>0 w2 (·; π) ⊆ lev>0 w2 (·; π0 ), π∈Π
and (13), we have
lev>0 w2 (·; π) ⊆ H.
π∈Π
Hence, we have that w2 ∈ W (Π2 ). (iv) Set Π = Π2 = cl H. Firstly, we prove that lev>0 s(·; π) ⊆ int H,
∀π ∈ Π.
(14)
In fact, for every π ∈ Π , (u, v) ∈ lev>0 s(·; π), by Proposition 2.1(iv), (v), we have (u, v) − π ∈ int H. Indeed, if (u, v) − π ∈ / int H, then (u, v) − π R
m+1
(15)
∈ ∂H ∪ int Hc ,
since
= int H ∪ ∂H ∪ int H . c
But, by Proposition 2.1(iv), if (u, v) − π ∈ ∂H, then s((u, v); π) = 0, which contradicts (u, v) ∈ lev>0 s(·; π). By Proposition 2.1(v), if (u, v) − π ∈ int Hc , then s((u, v); π) < 0, which contradicts (u, v) ∈ lev>0 s(·; π). Therefore, (15) is valid. From (15), we can obtain (u, v) ∈ int H + π ⊆ int H + cl H = int H. So, (14) is true. Secondly, we show that
lev>0 s(·; π) ⊆ int H.
(16)
π∈Π
In fact, let π0 = 0Rm+1 ∈ cl H. Then, by Proposition 2.1(iii)–(v), we have lev>0 s(·; π0 ) = int H. So, int H = lev>0 s(·; π0 ) ⊆
lev>0 s(·; π).
π∈Π
By (14), we have
lev>0 s(·; π) = int H.
π∈Π
It follows from (14) and (16) that the nonlinear function s is a strong separation function, i.e., the nonlinear function s ∈ S(Π2 ).
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4 Optimality Conditions In this section, we establish some weak and strong theorems of the alternative and then discuss the optimality conditions for the problem (1). Theorem 4.1 (i) The system (2) in the unknown x, and the system:
∃(θ, λ) ∈ Π1 s.t. θ f (x) ¯ − f (x) − ΔR+ λ, g(x) < 0,
∀x ∈ X, (17)
are not simultaneously possible. (ii) The system (2) in the unknown x, and the system: ∃π ∈ Π2 s.t. ΔH f (x) ¯ − f (x), g(x) + π > 0,
∀x ∈ X,
(18)
are not simultaneously possible. (iii) The system (2) in the unknown x, and the system: ∃π ∈ Π2 s.t. ΔH f (x) ¯ − f (x), g(x) − π ≥ 0,
∀x ∈ X,
(19)
are not simultaneously impossible. Proof From Proposition 3.1, weak and strong theorems of the alternative in [3], we know that the above results hold. Theorem 4.2 The system (2) in the unknown x, and the system:
∃(θ, λ) ∈ ]0, +∞[ ×D ∗ s.t. θ f (x) ¯ − f (x) − ΔR+ λ, g(x) ≤ 0,
∀x ∈ X, (20)
are not simultaneously possible. Proof The proof is similar to the one of Theorem 4.4.2 in [3], so we omit it here. By Theorems 4.1 and 4.2, we can obtain some sufficient and necessary optimality conditions for the problem (1). ¯ ∈ H∗ \{0Rm+1 }, such that Theorem 4.3 (i) Let x¯ ∈ R be given. If there exists (θ¯ , λ)
¯ g(x) < 0, ∀x ∈ X, (21) θ¯ f (x) ¯ − f (x) − ΔR+ λ, then x¯ is a global solution of the problem (1). ¯ ∈ H∗ with θ¯ > 0, such that (ii) Let x¯ ∈ R be given. If there exists (θ¯ , λ)
¯ g(x) ≤ 0, ∀x ∈ X, θ¯ f (x) ¯ − f (x) − ΔR+ λ,
(22)
then x¯ is a global solution of the problem (1). (iii) Let x¯ ∈ R be given. If there exists π ∈ cl H, such that ΔH f (x) ¯ − f (x), g(x) + π > 0, ∀x ∈ X,
(23)
then x¯ is a global solution of the problem (1). (iv) If x¯ is a global solution of the problem (1) and int H = ∅, then there exists π ∈ cl H, such that ΔH f (x) ¯ − f (x), g(x) − π ≥ 0, ∀x ∈ X. (24)
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Proof (i) (21) ⇒ the system (17) is possible. By Theorem 4.1(i), we know that the system (2) is impossible, i.e., Kx¯ ∩ H = ∅. Since x¯ ∈ R, x¯ is a global solution of the problem (1). (ii) (22) ⇒ the system (20) is possible. By Theorem 4.2, we know that the system (2) is impossible, i.e., Kx¯ ∩ H = ∅. Since x¯ ∈ R, x¯ is a global solution of the problem (1). (iii) (23) ⇒ the system (18) is possible. By Theorem 4.1(ii), we know that the system (2) is impossible, i.e., Kx¯ ∩ H = ∅. Since x¯ ∈ R, x¯ is a global solution of the problem (1). (iv) Since x¯ is a global solution of the problem (1), the system (2) is impossible. By Theorem 4.1(iii), we have that the system (19) is possible, namely, there exists π ∈ cl H, such that s Ax¯ (x); π = −ΔH f (x) ¯ − f (x), g(x) − π ≤ 0, ∀x ∈ X,
that is, (24) holds.
Now, we give Examples 4.1 and 4.2 to illustrate Theorem 4.3(ii) and (iv), respectively. Example 4.1 In the problem (1), set X := {(x1 , x2 ) ∈ R2 : x12 + x22 ≤ 1}, D = R+ , f (x) = x1 + 2x2 + 2, g(x) = 1√+ x1 +√ x2 . √ ¯ − f (x) = − 5 − x1 − 2x2 . Consider the vector x¯ = (− 55 , − 2 5 5 ) ∈ R, then f (x) Set θ¯ = 1 and λ¯ = 1, so we have √ w1 Ax¯ (x); θ¯ , λ¯ = − 5 − x1 − 2x2 − ΔR+ (1 + x1 + x2 ) √ = − 5 − x1 − 2x2 + x1 + x2 + 1 √ = − 5 + 1 − x2 ≤ 0, ∀(x1 , x2 ) ∈ X, √
√
that is, (22) is valid. We can compute that x¯ = (− 55 , − 2 5 5 ) is a global solution of the problem (1) by simplex method for linear programming. Example√4.2 In√this example, we shall continue to consider Example 4.1. We see that x¯ = (− 55 , − 2 5 5 ) is a global solution of the problem (1). There exists π = (2, 0) ∈ cl H, such that √ f (x) ¯ − f (x), g(x) − π = (− 5 − 2 − x1 − 2x2 , x1 + x2 + 1) ∈ / H, √ since − 5−2−x1 −2x2 < 0 for all (x1 , x2 ) ∈ X. By Proposition 2.1(iv), (v), we have ΔH f (x) ¯ − f (x), g(x) − π ≥ 0, ∀x ∈ X, which implies that (24) be valid. Using the nonlinear regular weak separation function in (3), we can get the following sufficient and necessary optimality condition for the problem (1). Theorem 4.4 Let θ¯ ∈ ]0, +∞[ be given. Let ¯ λ) := θ¯ u − ΔR+ λ, v , w3 (u, v; θ,
(u, v) ∈ R1+m , λ ∈ D ∗ .
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Then x¯ ∈ R is a global solution of the problem (1) if and only if
sup inf ∗ θ¯ f (x) ¯ − f (x) − ΔR+ λ, g(x) = 0.
(25)
Proof First, we prove ¯ λ) ⊇ lev>0 inf w3 (·; θ, ¯ λ) ⊇ H. H= lev>0 w3 (·; θ, ∗
(26)
x∈X λ∈D
λ∈D
λ∈D ∗
In fact, the first equality of (26) holds obviously, since function w3 is a regular weak separation function. The second inclusion of (26) is also true obviously. Now, we show that ¯ λ) ⊇ H. lev>0 inf ∗ w3 (·; θ, λ∈D
In fact, for all (u, v) ∈ H,
λ ∈ D∗ ,
by Proposition 2.1(iii), (iv), we have ΔR+ λ, v ≤ 0.
¯ λ) = θ¯ u > 0, i.e., Then, infλ∈D ∗ w3 (u, v; θ, ¯ λ) ⊇ H. lev>0 inf ∗ w3 (·; θ, λ∈D
From (26), we have ¯ λ). H = lev>0 inf ∗ w3 (·; θ,
(27)
λ∈D
If x¯ ∈ R is a global solution of the problem (1), then H ∩ Kx¯ = ∅. It follows from the equality (27) that ¯ λ) ∩ Kx¯ = ∅, lev>0 inf ∗ w3 (·; θ, λ∈D
which implies that
¯ − f (x) − ΔR+ λ, g(x) ≤ 0, inf θ¯ f (x) λ∈D ∗
Moreover,
∀x ∈ X.
inf ∗ θ¯ × 0 − ΔR+ λ, g(x) ¯ ≥ 0,
λ∈D
because
¯ λ). 0, g(x) ¯ ∈ cl H ⊆ lev≥0 inf ∗ w3 (·; θ, λ∈D
Therefore, one gets
¯ = 0, inf∗ θ¯ × 0 − ΔR+ λ, g(x)
λ∈D
which together with inequality (28), gives the equality (25). Conversely, suppose that equality (25) holds. Then one has
¯ − f (x) − ΔR+ λ, g(x) ≤ 0, ∀x ∈ X. inf∗ θ¯ f (x) λ∈D
(28)
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Simultaneously, we have ¯ λ) ⊇ H. lev>0 inf ∗ w3 (·; θ, λ∈D
So, the relation Kx¯ ∩ H = ∅ holds, which allows us to conclude that x¯ ∈ R is a global solution of the problem (1). The following example is given to illustrate Theorem 4.4. Example√ 4.3 In√ this example, we continue to consider Example 4.1. We know that x¯ = (− 55 , − 2 5 5 ) is a global solution of the problem (1) in Example 4.1. Let θ¯ ∈ ]0, +∞[ be given. By using direct computation, we can get ⎧ if x = x, ¯
⎨ 0, inf∗ θ¯ f (x) ¯ − f (x) − ΔR+ λ, g(x) = θ¯ (f (x) ¯ − f (x)), if x ∈ R\{x}, ¯ ⎩ λ∈D −∞, otherwise. So, supx∈X infλ∈D ∗ [θ¯ (f (x) ¯ − f (x)) − ΔR+ (λ, g(x))] = 0. On the other hand, if supx∈X infλ∈D ∗ [θ¯ (f (x) ¯ − f (x)) − ΔR+ (λ, g(x))] = 0, then
inf ∗ θ¯ f (x) ¯ − f (x) − ΔR+ λ, g(x) ≤ 0, ∀x ∈ X. λ∈D
In particular, let x ∈ R; we have
inf ∗ θ¯ f (x) ¯ − f (x) ≤ 0, ¯ − f (x) − ΔR+ λ, g(x) = θ¯ f (x) λ∈D
which implies that f (x) ¯ ≤ f (x),
∀x ∈ R.
The impossibility of the system (2) can be stated by means of separation arguments in the IS, viz, the sets Kx¯ and H of the IS lie in disjoint level sets of a suitable nonlinear function. Now, we introduce a concept related to the sets Kx¯ and H, and then obtain some optimality conditions of the problem (1). Definition 4.1 The sets Kx¯ and H admit a nonlinear separation, if and only if there ¯ ∈ H∗ \{0Rm+1 }, such that exists (θ¯ , λ)
¯ g(x) ≤ 0, x ∈ X. w1 Ax¯ (x); θ¯ , λ¯ = θ¯ f (x) ¯ − f (x) − ΔR+ λ, (29) Moreover, the separation is said to be regular iff θ¯ > 0 in (29). Proposition 4.1 If Kx¯ and H admit a regular nonlinear separation, then x¯ ∈ R is a global solution of the problem (1). Proof Since
(θ,λ)∈ ]0,+∞[×D ∗
lev>0 w1 (·; θ, λ) ⊇ H and (29) holds, we have Kx¯ ∩ H = ∅.
Hence, we can conclude that x¯ ∈ R is a global solution of the problem (1).
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In order to obtain a saddle-point sufficient optimality condition associated with the problem (1), we consider the following nonlinear function
L(x; θ, λ) := θf (x) + ΔR+ λ, g(x) , x ∈ X, (θ, λ) ∈ H∗ \{0Rm+1 }. (30) Theorem 4.5 The following statements are equivalent: (i) x¯ ∈ R, and Kx¯ and H admit a nonlinear separation; ¯ is a saddle point for L(x; θ, ¯ λ) ¯ ∈ H∗ \{0Rm+1 }, such that (x, ¯ λ) (ii) There exists (θ¯ , λ) ∗ on X × D , i.e., ¯ λ) ≤ L(x; L(x; ¯ θ, ¯ θ¯ , λ¯ ) ≤ L(x; θ¯ , λ¯ ),
∀x ∈ X, ∀λ ∈ D ∗ .
(31)
Proof (i) ⇒ (ii). Suppose that Kx¯ and H admit a nonlinear separation. Then there ¯ ∈ H∗ \{0Rm+1 } such that (29) holds. From (29), we can obtain that exists (θ¯ , λ)
¯ g(x) ΔR+ λ, ¯ ≥ 0. ¯ g(x)) ¯ = 0, since g(x) ¯ ∈ D. ThereBy Proposition 2.1(iii), (iv), we can see ΔR+ (λ, fore,
¯ g(x) ¯ λ) = θ¯ f (x) ¯ ≤ θ¯ f (x) ¯ + ΔR+ λ, ¯ L(x; ¯ θ, ¯ + ΔR+ λ, g(x) ¯ = L(x; ¯ θ¯ , λ), ∀λ ∈ D ∗ , since ΔR+ (λ, g(x)) ¯ ≤ 0 for all λ ∈ D ∗ . Now, we prove the second inequality of (31). In fact, since (29) holds, we have
¯ ∀x ∈ X. θ¯ f (x) + ΔR+ λ¯ , g(x) ≥ θ¯ f (x), ¯ = 0 that It follows from ΔR+ (λ¯ , g(x))
¯ g(x) ≥ θ¯ f (x) ¯ g(x) θ¯ f (x) + ΔR+ λ, ¯ + ΔR+ λ, ¯ ,
∀x ∈ X,
which is the second inequality of (31). ¯ be a saddle point for L(x; θ, ¯ λ) on X × D ∗ , that is, (ii) ⇒ (i). Suppose that (x, ¯ λ)
θ¯ f (x) ¯ + ΔR+ λ, g(x) ¯ ≤ θ¯ f (x) ¯ + ΔR+ λ¯ , g(x) ¯
≤ θ¯ f (x) + ΔR+ λ¯ , g(x) , ∀x ∈ X, ∀λ ∈ D ∗ . First of all, we prove that x¯ ∈ R. Ab absurdo, we suppose that g(x) ¯ ∈ / D. Then there ¯ < 0. Since D ∗ is a cone, we have exists λ∗ ∈ D ∗ , such that λ∗ , g(x)
tλ∗ ∈ D ∗ , ∀t ≥ 0 and t λ∗ , g(x) ¯ → −∞, t → +∞. By Proposition 2.1(v), (vi), we get
tΔR+ λ∗ , g(x) ¯ → +∞,
t → +∞,
which contradicts the first inequality. Taking λ = 0, it follows from the first inequality ¯ g(x)) ¯ g(x)) ¯ ≥ 0. Since x¯ ∈ R, ΔR+ (λ, ¯ ≤ 0 by Proposition 2.1(iii), that ΔR+ (λ, (iv). Therefore,
¯ g(x) ΔR+ λ, ¯ = 0. The second inequality coincides with (29) and the proof is complete.
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Remark 4.1 In the definition of the function L(x; θ, λ), we only need (θ, λ) ∈ H∗ \{0Rm+1 }. If we further assume θ > 0, then, without any loss of generality, we can set θ = 1. Since the function −ΔR+ (λ, g(x)) satisfies the properties of the function w(g(x); γ ) in (5.2.10.a) of [3], the nonlinear function L fulfills the hypotheses of Proposition 5.2.11 of [3]. So, in this case, Theorem 4.5 is a special case of Proposition 5.2.11. However, in the definition of the function L(x; θ, λ), θ may take 0. So, Theorem 4.5, in fact, is different from Proposition 5.2.11 of [3]. ¯ is a Corollary 4.1 Let x¯ ∈ X. If there exists θ¯ ∈ ]0, +∞[ and λ¯ ∈ D ∗ , such that (x, ¯ λ) ∗ ¯ saddle point for L(x; θ , λ) on X × D , then x¯ is a global solution of the problem (1). Proof From Theorem 4.5 and θ¯ ∈ ]0, +∞[, we have that the sets Kx¯ and H admit a regular nonlinear separation, and x¯ ∈ R. By Proposition 4.1, we obtain that x¯ is a global solution of the problem (1). Remark 4.2 The condition that θ¯ ∈ ]0, +∞[ in Corollary 4.1 is necessary, i.e., if there ¯ is a saddle point for L(x; θ¯ , λ) on ¯ λ) exists θ¯ = 0 and λ¯ ∈ D ∗ \{0Rm } such that (x, ∗ X × D , then x¯ ∈ X may be not a global solution of the problem (1). The following example is given to illustrate this case. Example 4.4 In the problem (1), set X = R, D = R2+ , f (x) = −3 + x, g1 (x) = −1 + x 2 , g2 (x) = x 2 (1 − x 2 ). ¯ with x¯ = 1, and λ¯ = (1, 1) is a saddle of L(x; 0, λ). Then we can get (x, ¯ λ) But x¯ = 1 is not a global solution of the problem (1). Hence, the condition that θ¯ ∈ ]0, +∞[ in Corollary 4.1 is necessary. ¯ ∈ H∗ \{0Rm+1 }. Assume that the Theorem 4.6 Let X be an open set in Rn and (θ¯ , λ) real valued function f (·) be locally Lipschitz on X. Suppose that the vector valued function g(·) be locally Lipschitz on X, i.e., for any x ∈ X, there exist a constant L and a neighborhood Nx of x, such that g(x1 ) − g(x2 ) ≤ Lx1 − x2 , ∀x1 , x2 ∈ Nx . ¯ is a saddle point for L(x; θ¯ , λ) on X × D ∗ , then If (x, ¯ λ) ⎧ ¯ λ), ¯ 0Rn ∈ ∂Lλ (x; ¯ θ, ⎪ ⎪ ⎪ ⎪ ⎨ 0 m ∈ ∂L (x; ¯ λ), ¯ x ¯ θ, R ¯ g(x)) ⎪ ¯ = 0, ΔR+ (λ, ⎪ ⎪ ⎪ ⎩ x¯ ∈ R. ¯ λ) on X × D ∗ , that is, Proof Suppose that (x, ¯ λ¯ ) be a saddle point for L(x; θ,
θ¯ f (x) ¯ + ΔR+ λ, g(x) ¯ ≤ θ¯ f (x) ¯ + ΔR+ λ¯ , g(x) ¯
≤ θ¯ f (x) + ΔR+ λ¯ , g(x) , ∀x ∈ X, ∀λ ∈ D ∗ . As in the proof of the sufficiency in Theorem 4.5, we can show that
¯ g(x) ¯ = 0. x¯ ∈ R and ΔR+ λ,
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¯ λ). We The first inequality implies that λ¯ be a global maximum solution of L(x; ¯ θ, now claim that the nonlinear function L(x; ¯ θ¯ , λ) is Lipschitz on D ∗ . In fact, for any λ1 , λ2 ∈ D ∗ , by Proposition 2.1(ii), we have
L(x; ¯ λ1 ) − L(x; ¯ λ2 ) = ΔR+ λ1 , g(x) ¯ θ, ¯ θ, ¯ − ΔR+ λ2 , g(x) ¯
≤ λ1 − λ2 , g(x) ¯ ¯ λ1 − λ2 . ≤ g(x) Hence, the nonlinear function L(x; ¯ θ¯ , λ) is Lipschitz on D ∗ . Therefore, by Lemma 2.1, one has 0Rm ∈ ∂Lλ (x; ¯ θ¯ , λ¯ ). ¯ We The second inequality implies that x¯ be a global minimum solution of L(x; θ¯ , λ). ¯ ¯ now claim that the nonlinear function L(x; θ, λ) is locally Lipschitz on X. Indeed, for any x ∈ X, since the functions f (·) and g(·) are locally Lipschitz on X, then there exists a neighborhood Nx of x, such that L(x1 ; θ, ¯ λ) ¯ − L(x2 ; θ¯ , λ) ¯
¯ g(x1 ) − ΔR+ λ, ¯ g(x2 ) = θ¯ f (x1 ) − θ¯ f (x2 ) + ΔR+ λ,
¯ g(x1 ) − ΔR+ λ, ¯ g(x2 ) ≤ θ¯ f (x1 ) − f (x2 ) + ΔR+ λ,
¯ g(x1 ) − g(x2 ) ≤ θ¯ L1 x1 − x2 + λ, ¯ ≤ θ¯ L1 x1 − x2 + L2 λx 1 − x2 = (θ¯ L1 + λ¯ L2 )x1 − x2 ,
∀x1 , x2 ∈ Nx ,
where L1 , L2 are Lipschitz constants for functions f (·) and g(·) at x, respectively. Hence, it follows from Lemma 2.1 that ¯ λ¯ ). 0Rn ∈ ∂Lx (x; ¯ θ, Thus, the proof is complete.
5 Concluding Remarks We have considered two nonlinear weak separation functions, a nonlinear regular weak separation function, and a nonlinear strong separation function by virtue of nonlinear scalarization function Δ, respectively. We have shown that the existence of a global saddle-point for a nonlinear function is equivalent to a nonlinear separation of two suitable subsets in the IS. Finally, we have proved some necessary and sufficient optimality conditions, in particular, a saddle-point sufficient optimality condition for the problem (1). Acknowledgements This research was supported by the National Natural Science Foundation of China (Grant Numbers: 10871216 and 11171362). The authors thank the two anonymous reviewers for their valuable comments and suggestions, which helped to improve the paper.
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