Solutions and optimality criteria for nonconvex quadratic-exponential
Math Meth Oper Res DOI 10.1007/s00186-007-0204-7 ORIGINAL ARTICLE
Solutions and optimality criteria for nonconvex quadratic-exponential minimization problem David Yang Gao · Ning Ruan
© Springer-Verlag 2008
Abstract This paper presents a set of complete solutions and optimality conditions for a nonconvex quadratic-exponential optimization problem. By using the canonical duality theory developed by the first author, the nonconvex primal problem in n-dimensional space can be converted into an one-dimensional canonical dual problem with zero duality gap, which can be solved easily to obtain all dual solutions. Each dual solution leads to a primal solution. Both global and local extremality conditions of these primal solutions can be identified by the triality theory associated with the canonical duality theory. Several examples are illustrated. Keywords Duality theory · Nonconvex programming · Global optimization · Quadratic-exponential function · Nonlinear algebraic equation · Triality 1 Primal problem and its dual form The primal problem to be solved is proposed as the following: 1 T T n (Pe ) min P(x) = x Ax − c x + W (x) : x ∈ R , 2
(1)
D. Y. Gao (B) · N. Ruan Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA e-mail: [email protected] N. Ruan School of Management, University of Shanghai for Science and Technology, Jungong Road, Shanghai 200093, China e-mail: [email protected]
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where A = A T ∈ Rn×n is a given indefinite matrix, c is a given vector in Rn , the nonconvex function W (x) is an exponential function with quadratic function exponent: 1 (2) W (x) = exp |Bx|2 − α , 2 where B ∈ Rm×n is a matrix, α > 0 is a positive constants, and |v| denotes the Euclidean norm of v. The nonconvex minimization problem (Pe ) frequently occurs in economics or statistics. Actually, the quadratic-exponential function can be used to model a large class of nonlinear phenomena (i.e. the so-called constitutive laws), such as plant and insect growth (Briggs et al. 1920), finite deformation elasticity (Gao and Ogden 2007), computational bio-chemistry (Floudas 2000, 2003), and bio-mechanics (Holzapfel and Ogden 2006). The criticality condition ∇ P(x) = 0 leads to a nonlinear equilibrium equation: 1 2 |Bx| − α B T Bx = c. (3) Ax + exp 2 Direct methods for solving this coupled nonlinear algebraic system are very difficult. Also the Eq. (3) is only a necessary condition for global minimizer of the problem (Pe ). Due to the nonconvexity of the target function P(x), the problem (Pe ) may possess many local minimizers. A general sufficient condition for identifying the global minimizer is a fundamental task in global optimization. Canonical duality theory developed in (Gao 2000a,b, 2003b, 2004) is a potentially powerful methodology for solving global optimization problems. This theory is composed mainly of a canonical dual transformation and a triality theory. The canonical dual transformation can be used to formulate perfect dual problems with zero duality gap, while the triality theory can be used to identify both local and global extrema. The purpose of this paper is to demonstrate a concrete application of this theory by solving the nonconvex primal problem (Pe ). We will show that by the use of the canonical dual transformation, the nonlinear coupled algebraic system in Rn can be converted into an algebraic equation in one-dimensional space. Therefore, a complete set of solutions are obtained. Both global and local minimizers can be identified by the triality theory. Following the standard procedure of the canonical dual transformation, we introduce a Gateaux ˆ differentiable geometrical operator ξ = (x) =
1 |Bx|2 − α 2
(4)
which is a quadratic map from Rn into Va = {ξ ∈ R | ξ ≥ −α}. Thus, the nonconvex function W (x) can be written in the canonical form W (x) = V ((x)),
(5)
where V (ξ ) = eξ is a canonical function on Va , i.e., the duality relation ς = ∇V (ξ ) = eξ
123
(6)
Complete solutions and optimality criteria for nonconvex quadratic-exponential minimization problem
is invertible for any given ξ ∈ Va [see the definition of the canonical function introduced in Gao (2000a)]. By letting U (x) = 21 x T Ax − c T x, the primal problem (Pe ) can be reformulated in the following canonical form: (P) min{P(x) = U (x) + V ((x)) : x ∈ Rn }.
(7)
Let Va∗ = {ς ∈ R | ς > 0} be the range of the duality mapping ς = ∇V (ξ ) : Va → Va∗ ⊂ R. So (ξ, ς ) forms a canonical duality pair on Va × Va∗ (cf. Gao 2000a) and the Legendre conjugate V ∗ can be uniquely defined by V ∗ (ς ) = sta{ξ ς − V (ξ ) : ξ ∈ R} = ς log ς − ς,
(8)
where sta{} denotes finding stationary points of the statement in {}. Thus, replacing W (x) = V ((x)) by (x)ς − V ∗ (ς ), the Gao-Strang complementary function (i.e. the so-called extended Lagrangian) (Gao 2000a, Gao and Strang 1989) can be defined by (x, ς ) = U (x) + (x)ς − V ∗ (ς ) 1 1 = x T Ax − c T x + ( |Bx|2 − α)ς − (ς log ς − ς ). 2 2
(9)
For a fixed ς , the criticality condition ∇x (x, ς ) = 0 leads to the following canonical equilibrium equation: Ax − c + ς B T Bx = 0.
(10)
Clearly, for any given ς > 0, if the vector c ∈ Col (A + ς B T B), i.e., c is in the collum space of (A + ς B T B), the general solution of the Eq. (10) is x = (A + ς B T B)+ c,
(11)
where A+ represents the Moore-Penrose generalized inverse of A. Substituting this result into the total complementary function , the canonical dual problem can be finally formulated as 1 (P d ) : sta P d (ς ) = − c T (A+ς B T B)+ c−(ς log ς −ς )−ας : ς ∈ Sa , 2 (12) where the dual feasible space is given by Sa = {ς ∈ R | ς > 0, c ∈ Col (Ad (ς ))},
(13)
and Ad (ς ) is defined by Ad (ς ) = A + ς B T B.
(14)
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Theorem 1 If ς¯ is a KKT point of (P d ), then the vector x¯ = A+ ¯ d (ς)c
(15)
is a critical point of (Pe ) and P(x) ¯ = P d (ς¯ ). Proof Suppose that ς¯ is a KKT point of (P d ), then we have ς¯ > 0, ∇ P d (ς¯ ) =
1 |B x| ¯ 2 − log ς¯ − α ≤ 0 2
ς¯ T ∇ P d (ς¯ ) = 0.
(16) (17)
By the fact that ς¯ > 0, the complementarity condition (17) leads to 1 |B x| ¯ 2 − log ς¯ − α = 0, 2 i.e., ς¯ = exp
1
¯ 2 |B x| x¯ =
2
− α . Thus, we have
A+ d (ς¯ )c
=
+ 1 2 T A + exp c |B x| ¯ −α B B 2
This shows that x¯ is a critical point of the primal problem (Pe ). ¯ we have Moreover, in term of x¯ = A+ d (ς)c, 1 P d (ς¯ ) = − c T A+ d (ς¯ )c − (ς¯ log ς¯ − ς¯ ) − α ς¯ 2 1 = x¯ T (A + ς¯ B T B)x¯ − c T x¯ − (ς¯ log ς¯ − ς) ¯ − α ς¯ 2 1 1 = x¯ T A x¯ − c T x¯ + ( |B x| ¯ 2 − α)ς¯ − (ς¯ log ς¯ − ς) ¯ 2 2 1 1 = x¯ T A x¯ − c T x¯ + ς¯ + ( |B x| ¯ 2 − log ς¯ − α)ς¯ 2 2 1 1 |B x| ¯ 2−α = x¯ T A x¯ − c T x¯ + exp 2 2 = P(x). ¯ This proves the theorem.
The next section will show that the global and local extrema of the function P : Rn → R only rely on critical points of the canonical dual function P d (ς ). 2 Global and local optimality criteria It is known that the criticality condition is only necessary for local minimization of the nonconvex problem (Pe ). In order to identify global and local extrema among the
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critical points of the problem (Pe ), we need to introduce some useful feasible spaces Sa+ = {ς ∈ Sa | Ad (ς ) 0},
(18)
= {ς ∈ Sa | Ad (ς ) ≺ 0}.
(19)
Sa−
Clearly, both Sa± are open convex subsets of R. By the canonical duality theory developed in Gao (2000a), we have the following result. Theorem 2 (Triality Theorem) Suppose that the vector ς¯ is a critical point of the ¯ canonical dual function P d (ς¯ ). Let x¯ = A+ d (ς)c. If ς¯ ∈ Sa+ , then ς¯ is a global maximizer of P d on Sa+ , the vector x¯ is a global minimizer of P on Rn , and P(x) ¯ = min P(x) = max P d (ς ) = P d (ς¯ ). x∈Rn
ς∈Sa+
(20)
If ς¯ ∈ Sa− , then on the neighborhood X0 × S0 ⊂ Rn × Sa− of (x, ¯ ς¯ ), we have that either P(x) ¯ = min P(x) = min P d (ς ) = P d (ς¯ )
(21)
P(x) ¯ = max P(x) = max P d (ς ) = P d (ς¯ ).
(22)
x∈X0
ς∈S0
holds, or x∈X0
ς∈S0
Proof By Theorem 1 and the canonical duality theory Gao (2000a, 2005), we know ¯ is that vector ς¯ ∈ Sa is a KKT point of the problem (P d ) if and only if x¯ = A+ d (ς)c a critical point of the problem (Pe ), and P(x) ¯ = (x, ¯ ς¯ ) = P d (ς¯ ). By the fact that the canonical dual function P d (ς ) is concave on Sa+ (which can be easily proved by ∇ 2 P d (ς ) < 0 ∀ς ∈ Sa+ ), the critical point ς¯ ∈ Sa+ is a global ¯ ς¯ ) is a saddle point of the total complementary maximizer of P d (ς ) over Sa+ , and (x, function (x, ς ) on Rn × Sa+ , i.e., is convex in x ∈ Rn and concave in ς ∈ Sa+ . Thus, by the well-known saddle Lagrangian duality theory (see Gao 2000a) we have P d (ς¯ ) = max P d (ς ) = max min (x, ς ) = min max (x, ς ) ς∈Sa+
ς∈Sa+ x∈Rn
x∈Rn ς∈Sa+
1 2 |Bx| − α ς − (ς log ς − ς ) = min 2 x∈Rn 1 T 1 x Ax − c T x + max ς + |Bx|2 − log ς − α ς = min 2 x∈Rn 2 ς∈Sa+ 1 T x Ax − c T x + max 2 ς∈Sa+
= min P(x) x∈Rn
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This proves the statement (20). ¯ is negative definite. In this case, the Gao-Strang If ς¯ ∈ Sa− , the matrix Ad (ς) complementary function (x, ¯ ς¯ ) is a so-called super-Lagrangian (seeGao (2000a)), i.e., it is locally concave in both x ∈ X0 ⊂ Rn and ς ∈ Sa− ⊂ Sa . Thus, by the triality theory developed in Gao (2000a), we have that either min max (x, ς ) = min max (x, ς )
(23)
max max (x, ς ) = max max (x, ς )
(24)
x∈X0 ς∈Sa−
ς∈S0 x∈Rn
or x∈X0 ς∈Sa−
ς∈S0 x∈Rn
holds on the neighborhood X0 × S0 of (x, ¯ ς¯ ). Thus, the equality (23) leads to the statement (21), while (24) leads to the statement (22). This proves the theorem. This Theorem shows that the extremality condition of the primal problem is control¯ ς) ¯ led by the critical points of the canonical dual problem, i.e., if ς¯ ∈ Sa+ , the vector x( ¯ ς) ¯ is a local minimizer (resp. maxiis a global minimizer of (Pe ); if ς¯ ∈ Sa− , then x( mizer) of (Pe ) if and only if the critical point ς¯ is a local minimizer (resp. maximizer) of P d on Sa− . In a special case when A is a diagonal matrix and B is an identity matrix, we have A+ d (ς ) =
1 ai + ς
.
(25)
In this case, 1 ci2 − (ς log ς − ς ) − ας. 2 ai + ς n
P d (ς ) = −
(26)
i=1
The criticality condition ∇ P d (ς ) = 0 gives the canonical dual algebraic equation: 2 n 1 ci − log ς − α = 0. 2 ai + ς
(27)
i=1
For the given α, {ci }, and {ai } such that a1 ≤ a2 ≤ · · · ≤ an , this dual algebraic equation (27) can be solved completely within each interval −ai+1 < ς < −ai such that ai < ai+1 (i = 1, 2, . . . , n). Similar problems and solutions were discussed in Gao (2003b, 2004). 3 Applications We now list a few examples to illustrate the applications of the theory presented in this paper.
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Complete solutions and optimality criteria for nonconvex quadratic-exponential minimization problem
6 4 2
-3
-2
-1
1
2
3
-2 -4 -6 Fig. 1 Graph of P(x) for one dimensional problem which has one global minimizer x1 = 2.36, one local minimizer x2 = −2.27, and one local maximizer x3 = −0.27
3.1 One-D nonconvex minimization First of all, let us consider one dimensional concave minimization problem:
1 2 1 x −2 : x ∈R . min P(x) = ax 2 − cx + exp 2 2
(28)
Sa = {ς ∈ R | ς > 0, a + ς = 0}.
(29)
1 P d (ς ) = − c2 /(a + ς ) − ς log ς − ς. 2
(30)
In this case,
The dual function is
If we choose c = 0.5, a = −2, the dual solution ς1 = 2.21 is a unique global maximizer of P d on Sa+ = {ς ∈ R+ | a + ς > 0}. It gives the global minimizer x1 = 2.36. It is easy to check that P(x1 ) = −4.56 = P d (ς1 ). While ς2 = 1.78 is a minimizer of P d on Sa− = {ς ∈ R+ | a + ς < 0}, which gives a local minimizer x2 = −2.27 and we have P(x2 ) = −2.24 = P d (ς2 ). And ς3 = 0.14 is a local maximizer of P d on Sa− = {ς ∈ R+ | a + ς < 0}, which gives a local maximizers x3 = −0.27 and we have P(x3 ) = 0.20 = P d (ς3 ). The graph of P(x) and P d (ς ) are shown in Figs. 1, 2.
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D. Y. Gao, N. Ruan 6 4 2 1
2
3
4
-2 -4 -6 -8 -10
Fig. 2 Graph of P d (ς ) for one dimensional problem which is concave on ς > 2 and has only one global maximizer ς1 = 2.21. But on the domain ς < 2, P d (ς ) is nonconvex which has one local minimizer ς2 = 1.78 and one local maximizer ς3 = 0.14
3.2 Two-D nonconvex minimization
1 min P(x1 , x2 ) = (a1 x12 + a2 x22 ) − c1 x1 − c2 x2 + 2 1 2 2 2 (x + x2 ) − 2 : x ∈ R . exp 2 1
(31)
On the dual feasible set Sa = {ς ∈ R2 | ς > 0, (a1 + ς )(a2 + ς ) = 0}. The canonical dual function has the form of
1 1 c1 d T a1 +ς P (ς ) = − [c1 , c2 ] − ς log ς − ς. 1 c2 2 a2 +ς
(32)
(33)
Case 1 a1 ≤ 0, a2 ≤ 0. We let c = (0.1, −0.3), a1 = −1, a2 = −1.2. The canonical dual problem has three critical points ς1 = 1.34 ∈ Sa+ = {ς ∈ R2 | ς > 1.2}, and ς2 = 0.94, ς3 = 0.14 ∈ Sa− = {ς ∈ R2 | 0 < ς < 1}. By Theorem 2, we know that x1 = {c1 /(a1 + ς1 ), c2 /(a2 + ς1 )}= {0.29, −2.12} is a global minimizer, x2 = {−1.60, 1.14} is a local minimizer, and x3 = {−0.12, 0.28} is
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Complete solutions and optimality criteria for nonconvex quadratic-exponential minimization problem 2
1
1
2
0
0
1
-1 -2 -2
0
-1
-1 -1 0 -2 -2
1 -2
-1.5
-1
-0.5
0
0.5
1
Fig. 3 Graphs of P(x) and its contour for two dimensional problem (Case I)
2 1
0.5
1
1.5
2
-1 -2 -3 -4 Fig. 4 Graph of P d (ς ) for two dimensional problem (Case I)
a local maximizer. It is easy to verify that P(x1 ) = P d (ς1 ) = −2.07 < P(x2 ) = P d (ς2 ) = −0.63 < P(x3 ) = P d (ς3 ) = 0.18 (see Figs. 3, 4). Case 2 a1 ≤ 0, a2 ≥ 0. We choose c = (0.1, −0.3), a1 = −1, a2 = 0.6. In this case, we have ς1 = 1.05 ∈ Sa+ = {ς ∈ R2 | ς > 1}, and ς2 = 0.95, ς3 = 0.15 ∈ Sa− = {ς ∈ R2 | 0 < ς < 1}.
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D. Y. Gao, N. Ruan 1.5
1
0.5
3 2 1 0 -1
1.5 1
0
0.5 -2
0
-0.5
-1 0
-0.5 1
-1 2
-1
-2
-1
1
0
2
Fig. 5 Graphs of P(x) and its contour for two dimensional problem (Case II) 1 0.5 0.5
1
1.5
2
-0.5 -1 -1.5 -2 -2.5 -3
Fig. 6
Graph of P d (ς ) for two dimensional problem (Case II)
thus, x1 ={2.02, −0.18} ∈ R2 is a global minimizer, x2 = {−1.96, −0.19} ∈ R2 is a local minimizer, and x3 = {−0.12, −0.40} ∈ R2 is a local maximizer. It is easy to verify that P(x1 ) = P d (ς1 ) = −1.23 < P(x2 ) = P d (ς2 ) = −0.83 < P(x3 ) = P d (ς3 ) = 0.08 (see Fig. 5, 6).
3.3 Two-D general nonconvex minimization We let A is a diagonal matrix and B is a 3 × 2 matrix such that the primal problem is 1 1 min P(x1 , x2 ) = (a1 x12 +a2 x22 )−c1 x1 −c2 x2 +exp |Bx|2 −4 : x ∈ R2 . 2 2 (34)
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Complete solutions and optimality criteria for nonconvex quadratic-exponential minimization problem
2
1 30 20
0 2
10 0 -10
1 -1 0
-2 -1
-1 -2
0 1
-2 2
-1
-2
0
1
2
Fig. 7 Graphs of P(x) and its contour for two dimensional general problem
c11 c12 , then on the dual feasible set B= c21 c22
Suppose
BT
Sa =
a1 + ς c11 ς c12 , ς ∈ R2 ς > 0, c ∈ Col ς c21 a2 + ς c22
(35)
the canonical dual function has the form of 1 P (ς ) = − [c1 , c2 ]T 2 d
a1 + ς c11 ς c12 ς c21 a2 + ς c22
+ ⎡
c1 − ς log ς − 3ς. (36) c2
−1 ⎣ −1 We let c = (0.5, −0.5), a1 = −2, a2 = 1.2, B = 2 points of the canonical dual problem are
⎤ −1 −2 ⎦. The two critical 1
ς1 = 0.94 ∈ Sa+ = {ς ∈ R2 | ς > 0.28}, and ς2 = 0.50 ∈ Sa− = {ς ∈ R2 | 0 < ς < 0.28}. By Theorem 2, we know that x1 = {2.03, −1.47} ∈ R2 is a global minimizer, and x2 = {−1.66, 0.87} ∈ R2 is a local minimizer. It is easy to verify that P(x1 ) = P d (ς1 ) = −3.65 < P(x2 ) = P d (ς2 ) = −0.53 (see Figs. 7, 8). Comparing Fig. 7 with Fig. 8 we can see clearly that the graph of the primal function is very flat, which implies a very slow convergent rate of any numerical method used for solving this problem directly. On the contrary, the dual problem with only one
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2 1 0.25
0.5
0.75
1
1.25
1.5
1.75
-1 -2 -3 -4 -5 -6 Fig. 8 Graph of P d (ς ) for two dimensional general problem
variable can be solved very easily to obtain all extreme points and the biggest dual solution ς1 leads to the global minimizer of the primal problem.
4 Conclusions In this paper we have presented a good application of the canonical duality theory to the nonconvex optimization problem (Pe ). Generally speaking, the nonconvex quadratic form with an exponential objective function can be used to model many nonconvex systems. By using the canonical dual transformation, the nonconvex primal problem in n-dimensional space can be converted into a one-dimensional canonical dual problem, which can be solved completely. Both global and local extrema can be identified by Theorem 2. As indicated in Gao (2000a, 2003b) that for any given nonconvex problem, as long as the geometrical operator (x) can be chosen properly and the canonical duality pairs can be identified correctly, the canonical dual transformation can be used to formulate perfect dual problems and the triality theory can be used to identify both global and local extrema. It is interesting to note that many different primal problems have the same canonical dual formula. In nonconvex analysis and finite deformation theory, the canonical dual formula is also called the pure complementary function and the related theory is known as the Gao principle (Li and Gupta 2006). This principle can be used to solve a class of nonconvex boundary value problems (Gao 1998, 2000c, Gao and Ogden 2007). In global optimization, extensive applications of the canonical duality theory have been given recently to solve some well known problems including concave minimization with inequality constraints (Gao 2005), polynomial minimization (Gao 2006), nonconvex minimization with box constraints (Gao 2007), quadratic minimization with general nonconvex constraints Gao et al. (2007), nonconvex fractional programming (Fang et al. 2007), integer programming (Fang et al. 2007, Wang et al. 2008), and minimization of general nonconvex functions of matrices (Ekeland and Gao 2007). Interested readers are refereed to the review articles (Gao 2003a, Gao and Sherali 2007).
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Solutions and optimality criteria for nonconvex quadratic-exponential minimization problem David Yang Gao · Ning Ruan
© Springer-Verlag 2008
Abstract This paper presents a set of complete solutions and optimality conditions for a nonconvex quadratic-exponential optimization problem. By using the canonical duality theory developed by the first author, the nonconvex primal problem in n-dimensional space can be converted into an one-dimensional canonical dual problem with zero duality gap, which can be solved easily to obtain all dual solutions. Each dual solution leads to a primal solution. Both global and local extremality conditions of these primal solutions can be identified by the triality theory associated with the canonical duality theory. Several examples are illustrated. Keywords Duality theory · Nonconvex programming · Global optimization · Quadratic-exponential function · Nonlinear algebraic equation · Triality 1 Primal problem and its dual form The primal problem to be solved is proposed as the following: 1 T T n (Pe ) min P(x) = x Ax − c x + W (x) : x ∈ R , 2
(1)
D. Y. Gao (B) · N. Ruan Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA e-mail: [email protected] N. Ruan School of Management, University of Shanghai for Science and Technology, Jungong Road, Shanghai 200093, China e-mail: [email protected]
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where A = A T ∈ Rn×n is a given indefinite matrix, c is a given vector in Rn , the nonconvex function W (x) is an exponential function with quadratic function exponent: 1 (2) W (x) = exp |Bx|2 − α , 2 where B ∈ Rm×n is a matrix, α > 0 is a positive constants, and |v| denotes the Euclidean norm of v. The nonconvex minimization problem (Pe ) frequently occurs in economics or statistics. Actually, the quadratic-exponential function can be used to model a large class of nonlinear phenomena (i.e. the so-called constitutive laws), such as plant and insect growth (Briggs et al. 1920), finite deformation elasticity (Gao and Ogden 2007), computational bio-chemistry (Floudas 2000, 2003), and bio-mechanics (Holzapfel and Ogden 2006). The criticality condition ∇ P(x) = 0 leads to a nonlinear equilibrium equation: 1 2 |Bx| − α B T Bx = c. (3) Ax + exp 2 Direct methods for solving this coupled nonlinear algebraic system are very difficult. Also the Eq. (3) is only a necessary condition for global minimizer of the problem (Pe ). Due to the nonconvexity of the target function P(x), the problem (Pe ) may possess many local minimizers. A general sufficient condition for identifying the global minimizer is a fundamental task in global optimization. Canonical duality theory developed in (Gao 2000a,b, 2003b, 2004) is a potentially powerful methodology for solving global optimization problems. This theory is composed mainly of a canonical dual transformation and a triality theory. The canonical dual transformation can be used to formulate perfect dual problems with zero duality gap, while the triality theory can be used to identify both local and global extrema. The purpose of this paper is to demonstrate a concrete application of this theory by solving the nonconvex primal problem (Pe ). We will show that by the use of the canonical dual transformation, the nonlinear coupled algebraic system in Rn can be converted into an algebraic equation in one-dimensional space. Therefore, a complete set of solutions are obtained. Both global and local minimizers can be identified by the triality theory. Following the standard procedure of the canonical dual transformation, we introduce a Gateaux ˆ differentiable geometrical operator ξ = (x) =
1 |Bx|2 − α 2
(4)
which is a quadratic map from Rn into Va = {ξ ∈ R | ξ ≥ −α}. Thus, the nonconvex function W (x) can be written in the canonical form W (x) = V ((x)),
(5)
where V (ξ ) = eξ is a canonical function on Va , i.e., the duality relation ς = ∇V (ξ ) = eξ
123
(6)
Complete solutions and optimality criteria for nonconvex quadratic-exponential minimization problem
is invertible for any given ξ ∈ Va [see the definition of the canonical function introduced in Gao (2000a)]. By letting U (x) = 21 x T Ax − c T x, the primal problem (Pe ) can be reformulated in the following canonical form: (P) min{P(x) = U (x) + V ((x)) : x ∈ Rn }.
(7)
Let Va∗ = {ς ∈ R | ς > 0} be the range of the duality mapping ς = ∇V (ξ ) : Va → Va∗ ⊂ R. So (ξ, ς ) forms a canonical duality pair on Va × Va∗ (cf. Gao 2000a) and the Legendre conjugate V ∗ can be uniquely defined by V ∗ (ς ) = sta{ξ ς − V (ξ ) : ξ ∈ R} = ς log ς − ς,
(8)
where sta{} denotes finding stationary points of the statement in {}. Thus, replacing W (x) = V ((x)) by (x)ς − V ∗ (ς ), the Gao-Strang complementary function (i.e. the so-called extended Lagrangian) (Gao 2000a, Gao and Strang 1989) can be defined by (x, ς ) = U (x) + (x)ς − V ∗ (ς ) 1 1 = x T Ax − c T x + ( |Bx|2 − α)ς − (ς log ς − ς ). 2 2
(9)
For a fixed ς , the criticality condition ∇x (x, ς ) = 0 leads to the following canonical equilibrium equation: Ax − c + ς B T Bx = 0.
(10)
Clearly, for any given ς > 0, if the vector c ∈ Col (A + ς B T B), i.e., c is in the collum space of (A + ς B T B), the general solution of the Eq. (10) is x = (A + ς B T B)+ c,
(11)
where A+ represents the Moore-Penrose generalized inverse of A. Substituting this result into the total complementary function , the canonical dual problem can be finally formulated as 1 (P d ) : sta P d (ς ) = − c T (A+ς B T B)+ c−(ς log ς −ς )−ας : ς ∈ Sa , 2 (12) where the dual feasible space is given by Sa = {ς ∈ R | ς > 0, c ∈ Col (Ad (ς ))},
(13)
and Ad (ς ) is defined by Ad (ς ) = A + ς B T B.
(14)
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Theorem 1 If ς¯ is a KKT point of (P d ), then the vector x¯ = A+ ¯ d (ς)c
(15)
is a critical point of (Pe ) and P(x) ¯ = P d (ς¯ ). Proof Suppose that ς¯ is a KKT point of (P d ), then we have ς¯ > 0, ∇ P d (ς¯ ) =
1 |B x| ¯ 2 − log ς¯ − α ≤ 0 2
ς¯ T ∇ P d (ς¯ ) = 0.
(16) (17)
By the fact that ς¯ > 0, the complementarity condition (17) leads to 1 |B x| ¯ 2 − log ς¯ − α = 0, 2 i.e., ς¯ = exp
1
¯ 2 |B x| x¯ =
2
− α . Thus, we have
A+ d (ς¯ )c
=
+ 1 2 T A + exp c |B x| ¯ −α B B 2
This shows that x¯ is a critical point of the primal problem (Pe ). ¯ we have Moreover, in term of x¯ = A+ d (ς)c, 1 P d (ς¯ ) = − c T A+ d (ς¯ )c − (ς¯ log ς¯ − ς¯ ) − α ς¯ 2 1 = x¯ T (A + ς¯ B T B)x¯ − c T x¯ − (ς¯ log ς¯ − ς) ¯ − α ς¯ 2 1 1 = x¯ T A x¯ − c T x¯ + ( |B x| ¯ 2 − α)ς¯ − (ς¯ log ς¯ − ς) ¯ 2 2 1 1 = x¯ T A x¯ − c T x¯ + ς¯ + ( |B x| ¯ 2 − log ς¯ − α)ς¯ 2 2 1 1 |B x| ¯ 2−α = x¯ T A x¯ − c T x¯ + exp 2 2 = P(x). ¯ This proves the theorem.
The next section will show that the global and local extrema of the function P : Rn → R only rely on critical points of the canonical dual function P d (ς ). 2 Global and local optimality criteria It is known that the criticality condition is only necessary for local minimization of the nonconvex problem (Pe ). In order to identify global and local extrema among the
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Complete solutions and optimality criteria for nonconvex quadratic-exponential minimization problem
critical points of the problem (Pe ), we need to introduce some useful feasible spaces Sa+ = {ς ∈ Sa | Ad (ς ) 0},
(18)
= {ς ∈ Sa | Ad (ς ) ≺ 0}.
(19)
Sa−
Clearly, both Sa± are open convex subsets of R. By the canonical duality theory developed in Gao (2000a), we have the following result. Theorem 2 (Triality Theorem) Suppose that the vector ς¯ is a critical point of the ¯ canonical dual function P d (ς¯ ). Let x¯ = A+ d (ς)c. If ς¯ ∈ Sa+ , then ς¯ is a global maximizer of P d on Sa+ , the vector x¯ is a global minimizer of P on Rn , and P(x) ¯ = min P(x) = max P d (ς ) = P d (ς¯ ). x∈Rn
ς∈Sa+
(20)
If ς¯ ∈ Sa− , then on the neighborhood X0 × S0 ⊂ Rn × Sa− of (x, ¯ ς¯ ), we have that either P(x) ¯ = min P(x) = min P d (ς ) = P d (ς¯ )
(21)
P(x) ¯ = max P(x) = max P d (ς ) = P d (ς¯ ).
(22)
x∈X0
ς∈S0
holds, or x∈X0
ς∈S0
Proof By Theorem 1 and the canonical duality theory Gao (2000a, 2005), we know ¯ is that vector ς¯ ∈ Sa is a KKT point of the problem (P d ) if and only if x¯ = A+ d (ς)c a critical point of the problem (Pe ), and P(x) ¯ = (x, ¯ ς¯ ) = P d (ς¯ ). By the fact that the canonical dual function P d (ς ) is concave on Sa+ (which can be easily proved by ∇ 2 P d (ς ) < 0 ∀ς ∈ Sa+ ), the critical point ς¯ ∈ Sa+ is a global ¯ ς¯ ) is a saddle point of the total complementary maximizer of P d (ς ) over Sa+ , and (x, function (x, ς ) on Rn × Sa+ , i.e., is convex in x ∈ Rn and concave in ς ∈ Sa+ . Thus, by the well-known saddle Lagrangian duality theory (see Gao 2000a) we have P d (ς¯ ) = max P d (ς ) = max min (x, ς ) = min max (x, ς ) ς∈Sa+
ς∈Sa+ x∈Rn
x∈Rn ς∈Sa+
1 2 |Bx| − α ς − (ς log ς − ς ) = min 2 x∈Rn 1 T 1 x Ax − c T x + max ς + |Bx|2 − log ς − α ς = min 2 x∈Rn 2 ς∈Sa+ 1 T x Ax − c T x + max 2 ς∈Sa+
= min P(x) x∈Rn
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D. Y. Gao, N. Ruan
This proves the statement (20). ¯ is negative definite. In this case, the Gao-Strang If ς¯ ∈ Sa− , the matrix Ad (ς) complementary function (x, ¯ ς¯ ) is a so-called super-Lagrangian (seeGao (2000a)), i.e., it is locally concave in both x ∈ X0 ⊂ Rn and ς ∈ Sa− ⊂ Sa . Thus, by the triality theory developed in Gao (2000a), we have that either min max (x, ς ) = min max (x, ς )
(23)
max max (x, ς ) = max max (x, ς )
(24)
x∈X0 ς∈Sa−
ς∈S0 x∈Rn
or x∈X0 ς∈Sa−
ς∈S0 x∈Rn
holds on the neighborhood X0 × S0 of (x, ¯ ς¯ ). Thus, the equality (23) leads to the statement (21), while (24) leads to the statement (22). This proves the theorem. This Theorem shows that the extremality condition of the primal problem is control¯ ς) ¯ led by the critical points of the canonical dual problem, i.e., if ς¯ ∈ Sa+ , the vector x( ¯ ς) ¯ is a local minimizer (resp. maxiis a global minimizer of (Pe ); if ς¯ ∈ Sa− , then x( mizer) of (Pe ) if and only if the critical point ς¯ is a local minimizer (resp. maximizer) of P d on Sa− . In a special case when A is a diagonal matrix and B is an identity matrix, we have A+ d (ς ) =
1 ai + ς
.
(25)
In this case, 1 ci2 − (ς log ς − ς ) − ας. 2 ai + ς n
P d (ς ) = −
(26)
i=1
The criticality condition ∇ P d (ς ) = 0 gives the canonical dual algebraic equation: 2 n 1 ci − log ς − α = 0. 2 ai + ς
(27)
i=1
For the given α, {ci }, and {ai } such that a1 ≤ a2 ≤ · · · ≤ an , this dual algebraic equation (27) can be solved completely within each interval −ai+1 < ς < −ai such that ai < ai+1 (i = 1, 2, . . . , n). Similar problems and solutions were discussed in Gao (2003b, 2004). 3 Applications We now list a few examples to illustrate the applications of the theory presented in this paper.
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Complete solutions and optimality criteria for nonconvex quadratic-exponential minimization problem
6 4 2
-3
-2
-1
1
2
3
-2 -4 -6 Fig. 1 Graph of P(x) for one dimensional problem which has one global minimizer x1 = 2.36, one local minimizer x2 = −2.27, and one local maximizer x3 = −0.27
3.1 One-D nonconvex minimization First of all, let us consider one dimensional concave minimization problem:
1 2 1 x −2 : x ∈R . min P(x) = ax 2 − cx + exp 2 2
(28)
Sa = {ς ∈ R | ς > 0, a + ς = 0}.
(29)
1 P d (ς ) = − c2 /(a + ς ) − ς log ς − ς. 2
(30)
In this case,
The dual function is
If we choose c = 0.5, a = −2, the dual solution ς1 = 2.21 is a unique global maximizer of P d on Sa+ = {ς ∈ R+ | a + ς > 0}. It gives the global minimizer x1 = 2.36. It is easy to check that P(x1 ) = −4.56 = P d (ς1 ). While ς2 = 1.78 is a minimizer of P d on Sa− = {ς ∈ R+ | a + ς < 0}, which gives a local minimizer x2 = −2.27 and we have P(x2 ) = −2.24 = P d (ς2 ). And ς3 = 0.14 is a local maximizer of P d on Sa− = {ς ∈ R+ | a + ς < 0}, which gives a local maximizers x3 = −0.27 and we have P(x3 ) = 0.20 = P d (ς3 ). The graph of P(x) and P d (ς ) are shown in Figs. 1, 2.
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D. Y. Gao, N. Ruan 6 4 2 1
2
3
4
-2 -4 -6 -8 -10
Fig. 2 Graph of P d (ς ) for one dimensional problem which is concave on ς > 2 and has only one global maximizer ς1 = 2.21. But on the domain ς < 2, P d (ς ) is nonconvex which has one local minimizer ς2 = 1.78 and one local maximizer ς3 = 0.14
3.2 Two-D nonconvex minimization
1 min P(x1 , x2 ) = (a1 x12 + a2 x22 ) − c1 x1 − c2 x2 + 2 1 2 2 2 (x + x2 ) − 2 : x ∈ R . exp 2 1
(31)
On the dual feasible set Sa = {ς ∈ R2 | ς > 0, (a1 + ς )(a2 + ς ) = 0}. The canonical dual function has the form of
1 1 c1 d T a1 +ς P (ς ) = − [c1 , c2 ] − ς log ς − ς. 1 c2 2 a2 +ς
(32)
(33)
Case 1 a1 ≤ 0, a2 ≤ 0. We let c = (0.1, −0.3), a1 = −1, a2 = −1.2. The canonical dual problem has three critical points ς1 = 1.34 ∈ Sa+ = {ς ∈ R2 | ς > 1.2}, and ς2 = 0.94, ς3 = 0.14 ∈ Sa− = {ς ∈ R2 | 0 < ς < 1}. By Theorem 2, we know that x1 = {c1 /(a1 + ς1 ), c2 /(a2 + ς1 )}= {0.29, −2.12} is a global minimizer, x2 = {−1.60, 1.14} is a local minimizer, and x3 = {−0.12, 0.28} is
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Complete solutions and optimality criteria for nonconvex quadratic-exponential minimization problem 2
1
1
2
0
0
1
-1 -2 -2
0
-1
-1 -1 0 -2 -2
1 -2
-1.5
-1
-0.5
0
0.5
1
Fig. 3 Graphs of P(x) and its contour for two dimensional problem (Case I)
2 1
0.5
1
1.5
2
-1 -2 -3 -4 Fig. 4 Graph of P d (ς ) for two dimensional problem (Case I)
a local maximizer. It is easy to verify that P(x1 ) = P d (ς1 ) = −2.07 < P(x2 ) = P d (ς2 ) = −0.63 < P(x3 ) = P d (ς3 ) = 0.18 (see Figs. 3, 4). Case 2 a1 ≤ 0, a2 ≥ 0. We choose c = (0.1, −0.3), a1 = −1, a2 = 0.6. In this case, we have ς1 = 1.05 ∈ Sa+ = {ς ∈ R2 | ς > 1}, and ς2 = 0.95, ς3 = 0.15 ∈ Sa− = {ς ∈ R2 | 0 < ς < 1}.
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D. Y. Gao, N. Ruan 1.5
1
0.5
3 2 1 0 -1
1.5 1
0
0.5 -2
0
-0.5
-1 0
-0.5 1
-1 2
-1
-2
-1
1
0
2
Fig. 5 Graphs of P(x) and its contour for two dimensional problem (Case II) 1 0.5 0.5
1
1.5
2
-0.5 -1 -1.5 -2 -2.5 -3
Fig. 6
Graph of P d (ς ) for two dimensional problem (Case II)
thus, x1 ={2.02, −0.18} ∈ R2 is a global minimizer, x2 = {−1.96, −0.19} ∈ R2 is a local minimizer, and x3 = {−0.12, −0.40} ∈ R2 is a local maximizer. It is easy to verify that P(x1 ) = P d (ς1 ) = −1.23 < P(x2 ) = P d (ς2 ) = −0.83 < P(x3 ) = P d (ς3 ) = 0.08 (see Fig. 5, 6).
3.3 Two-D general nonconvex minimization We let A is a diagonal matrix and B is a 3 × 2 matrix such that the primal problem is 1 1 min P(x1 , x2 ) = (a1 x12 +a2 x22 )−c1 x1 −c2 x2 +exp |Bx|2 −4 : x ∈ R2 . 2 2 (34)
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Complete solutions and optimality criteria for nonconvex quadratic-exponential minimization problem
2
1 30 20
0 2
10 0 -10
1 -1 0
-2 -1
-1 -2
0 1
-2 2
-1
-2
0
1
2
Fig. 7 Graphs of P(x) and its contour for two dimensional general problem
c11 c12 , then on the dual feasible set B= c21 c22
Suppose
BT
Sa =
a1 + ς c11 ς c12 , ς ∈ R2 ς > 0, c ∈ Col ς c21 a2 + ς c22
(35)
the canonical dual function has the form of 1 P (ς ) = − [c1 , c2 ]T 2 d
a1 + ς c11 ς c12 ς c21 a2 + ς c22
+ ⎡
c1 − ς log ς − 3ς. (36) c2
−1 ⎣ −1 We let c = (0.5, −0.5), a1 = −2, a2 = 1.2, B = 2 points of the canonical dual problem are
⎤ −1 −2 ⎦. The two critical 1
ς1 = 0.94 ∈ Sa+ = {ς ∈ R2 | ς > 0.28}, and ς2 = 0.50 ∈ Sa− = {ς ∈ R2 | 0 < ς < 0.28}. By Theorem 2, we know that x1 = {2.03, −1.47} ∈ R2 is a global minimizer, and x2 = {−1.66, 0.87} ∈ R2 is a local minimizer. It is easy to verify that P(x1 ) = P d (ς1 ) = −3.65 < P(x2 ) = P d (ς2 ) = −0.53 (see Figs. 7, 8). Comparing Fig. 7 with Fig. 8 we can see clearly that the graph of the primal function is very flat, which implies a very slow convergent rate of any numerical method used for solving this problem directly. On the contrary, the dual problem with only one
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D. Y. Gao, N. Ruan
2 1 0.25
0.5
0.75
1
1.25
1.5
1.75
-1 -2 -3 -4 -5 -6 Fig. 8 Graph of P d (ς ) for two dimensional general problem
variable can be solved very easily to obtain all extreme points and the biggest dual solution ς1 leads to the global minimizer of the primal problem.
4 Conclusions In this paper we have presented a good application of the canonical duality theory to the nonconvex optimization problem (Pe ). Generally speaking, the nonconvex quadratic form with an exponential objective function can be used to model many nonconvex systems. By using the canonical dual transformation, the nonconvex primal problem in n-dimensional space can be converted into a one-dimensional canonical dual problem, which can be solved completely. Both global and local extrema can be identified by Theorem 2. As indicated in Gao (2000a, 2003b) that for any given nonconvex problem, as long as the geometrical operator (x) can be chosen properly and the canonical duality pairs can be identified correctly, the canonical dual transformation can be used to formulate perfect dual problems and the triality theory can be used to identify both global and local extrema. It is interesting to note that many different primal problems have the same canonical dual formula. In nonconvex analysis and finite deformation theory, the canonical dual formula is also called the pure complementary function and the related theory is known as the Gao principle (Li and Gupta 2006). This principle can be used to solve a class of nonconvex boundary value problems (Gao 1998, 2000c, Gao and Ogden 2007). In global optimization, extensive applications of the canonical duality theory have been given recently to solve some well known problems including concave minimization with inequality constraints (Gao 2005), polynomial minimization (Gao 2006), nonconvex minimization with box constraints (Gao 2007), quadratic minimization with general nonconvex constraints Gao et al. (2007), nonconvex fractional programming (Fang et al. 2007), integer programming (Fang et al. 2007, Wang et al. 2008), and minimization of general nonconvex functions of matrices (Ekeland and Gao 2007). Interested readers are refereed to the review articles (Gao 2003a, Gao and Sherali 2007).
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