Solutions and optimality criteria for nonconvex ... - Semantic Scholar

J Glob Optim DOI 10.1007/s10898-009-9399-x

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Solutions and optimality criteria for nonconvex constrained global optimization problems with connections between canonical and Lagrangian duality David Yang Gao · Ning Ruan · Hanif D. Sherali

Received: 2 November 2007 / Accepted: 6 January 2009 © Springer Science+Business Media, LLC. 2009

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Abstract This paper presents a canonical duality theory for solving a general nonconvex quadratic minimization problem with nonconvex constraints. By using the canonical dual transformation developed by the first author, the nonconvex primal problem can be converted into a canonical dual problem with zero duality gap. A general analytical solution form is obtained. Both global and local extrema of the nonconvex problem can be identified by the triality theory associated with the canonical duality theory. Illustrative applications to quadratic minimization with multiple quadratic constraints, box/integer constraints, and general nonconvex polynomial constraints are discussed, along with insightful connections to classical Lagrangian duality. Criteria for the existence and uniqueness of optimal solutions are presented. Several numerical examples are provided.

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Keywords Canonical duality theory · Triality · Lagrangian duality · Global optimization · Integer programming

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1 Introduction

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In this paper, we address the following general constrained nonlinear programming problem:
 1 (P ) : min P(x) = x T Ax − x T f : x ∈ Xc , (1) 2 where A = {Ai j } ∈ Rn×n is a symmetric matrix, f ∈ Rn is a given vector, the feasible space Xk ⊂ Rn is defined as   (2) Xc = x ∈ Xa | g(x) ≤ d ∈ Rm , D. Y. Gao (B) · N. Ruan Department of Mathematics, Virginia Tech, Blacksburg, VA 24061, USA e-mail: [email protected]

D. Y. Gao · N. Ruan · H. D. Sherali Grado Department of Industrial and Systems Engineering, Virginia Tech, Blacksburg, VA 24061, USA

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L(x, σ ) =

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1 T x Ax − x T f + σ T (g(x) − d). 2

P ∗ (σ ) = inf L(x, σ ),

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(4)

x∈Xa

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where, under certain constraint qualifications that insure the existence of a Karush– Kuhn–Tucker (KKT) solution (see [1]), we have the following strong min–max duality relation: inf P(x) = sup P ∗ (σ ). σ ∈Rm+

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x∈Xc

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inf P(x) ≥ sup P ∗ (σ ). σ ∈Rm+

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(5)

In this case, the problem can be solved easily by any well-developed convex programming technique. However, due to the assumed nonconvexity of Problem (P ), the Lagrangian L(x, σ ) is no longer a saddle function and the Fenchel-Young inequality leads to only the following weak duality relation in general:

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(3)

If all the components of g(x) are convex functions, and A  0, i.e., positive semidefinite (PSD), then Problem (P ) has a convex quadratic objective function and convex constraints, and the Lagrangian is a saddle function, i.e., L(x, σ ) is convex in the primal variables x, concave (linear) in the dual variables (Lagrange multipliers) σ , and the Lagrangian dual problem can be easily defined by the Fenchel–Moreau–Rockafellar transformation

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where g(x) = {gα (x)} : Xa → Rm is a given vector-valued differentiable (not necessary convex) function, Xa is a convex open set in Rn , and d ∈ Rm is a given vector. (We follow the traditional tensor notation used in finite deformation theory [10], where the indices i, j represent components of entities in Rn or Rn×n , while α, β represent components in other spaces). The problem (P ) involves minimizing a nonconvex quadratic function over a nonconvex m feasible space. By introducing a Lagrangian multiplier vector σ ∈ Rm + = {σ ∈ R | σ ≥ 0} m to relax the inequality constraints in Xc , the classical Lagrangian L : Xa × R+ → R is given by

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(6)

The slack θ in the inequality (6) is called the duality gap in global optimization, where possibly, θ = ∞. This duality gap shows that the well-developed Fenchel–Moreau–Rockafellar duality theory can be used only for solving convex minimization problems. Also, due to the nonconvexity of the objective function and/or constraints, the problem may have multiple local solutions. The identification of a global minimizer has been a fundamentally challenging task in global optimization. The classical Lagrangian duality theory was originally developed in the context of analytical mechanics [38]. In the realm of linear elasticity, the primal problem (P ) is usually a convex variational problem in functional space, while its dual (P ∗ ) is called the complementary variational problem. In mechanics and mathematical physics, the terminology “complementary” connotes the strong (or perfect) duality, i.e., the so-called canonical duality [10], and the strong duality relation (5) is also called the complementary variational principle. However, in finite deformation elasticity, the primal function P(x) is usually nonconvex and the existence of a purely stress-(dual variable)-based complementary variational principle has invoked a lively debate existing for more than 50 years, since E. Reissner (1953) [37,39,41].

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2 Canonical dual problem and strong duality theorem

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In order to close the duality gap inherent in the classical Lagrange duality theory, a socalled canonical duality theory has been developed, first in nonconvex analysis [27] and mechanics [7,10], and then in global optimization [12,16,26]. This new theory is composed mainly of a potentially useful canonical dual transformation, an associated complementarydual principle, and a triality theory, whose components comprise a saddle min–max duality and two pairs of double-min, double-max dualities. The canonical dual transformation can be used to formulate perfect dual problems without a duality gap; the complementary-dual principle shows that the primal problem is equivalent to its canonical dual in the sense that they have the same KKT points; while the triality theory can be used to identify both global and local extrema. The key step in the canonical dual transformation is to introduce a geometrical operator ξ = (x) and a canonical function V (ξ ) (whose Legendre or Fenchel conjugate can be uniquely defined) such that the nonconvex constraint can be written in the canonical form g(x) = V ((x)). Thus, the complementary (i.e., the canonical dual) function can be uniquely formulated via the Legendre-Fenchel transformation. The original idea of this canonical dual transformation comes from the joint work of Gao and Strang [27] on nonconvex/nonsmooth variational problems. They discovered that a necessary condition for a minimizer depends on the Gâteaux derivative t of the geometrical operator, while a sufficient condition and the so-called complementary gap function depends on the complementary operator c (x) := (x) − t x. This complementary gap function closes the duality gap present in Lagrangian duality and plays an important role in nonconvex analysis and global optimization. Some open problems in one-dimensional nonlinear elasticity have thus been solved recently [22,23]. A comprehensive review on the canonical duality theory and connections between nonconvex mechanics and global optimization appear in Gao and Sherali [26]. The purpose of the present paper is to illustrate the application of the canonical duality theory for solving the foregoing quadratic minimization problem with nonconvex constraints. In the next section, we will show how to use the canonical dual transformation to convert the nonconvex constrained problem into a canonical dual problem, in order to derive related global optimality conditions. Certain particular cases along with numerical examples are used to illustrate the theory in Sect. 3 (quadratic constraints), Sect. 4 (box and integer constraints), and Sect. 5 (polynomial constraints), for using the canonical dual problem to obtain all critical points. We also expose certain insightful connections with classical Lagrangian duality for these special applications. Finally, Sect. 6 provides a summary, some open problems, and conclusions.

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For analytical convenience, we introduce an indicator function of the feasible set Xc :
0 if ǫ ≤ d W (ǫ) = +∞ otherwise and let

(7)

1 U (x) ≡ −P(x) = x T f − x T Ax. 2

Then the primal problem (P ) can be written in the following unconstrained form: min {(x) = W (g(x)) − U (x) : x ∈ Xa } .

(8)

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which is convex and lower semi-continuous (l.s.c.) on Rm . From convex analysis [2,44], the following relations hold for (ǫ, σ ) ∈ Rm × Rm : σ ∈ ∂ W (ǫ) ⇔ ǫ ∈ ∂ W ♯ (σ ) ⇔ W (ǫ) + W ♯ (σ ) = ǫ T σ .

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Replacing W (g(x)) in (x) by g T (x)σ − W ♯ (σ ) based on the Fenchel–Young equality, the extended Lagrangian o : Xa × Rm → R ∪ {∞} associated with the problem (8) can be given as

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o (x, σ ) = g T (x)σ − W ♯ (σ ) − U (x).

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ξ = {ξβα } = (x) : Xa ⊂ Rn → Ea ⊂ Rm× pα ,

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and a canonical function V : Ea → written in the canonical form:

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(11)

such that the nonconvex constraint g(x) can be

g(x) = V ((x)).

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(10)

Clearly, we have o (x, σ ) = L(x, σ ), ∀(x, σ ) ∈ Xa × Rm +. Since g(x) is a nonconvex function, following the standard procedure of the canonical dual transformation (see [10]), we assume that there exists a Gâteaux differentiable geometrical operator

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By the Fenchel transformation, the conjugate function W ♯ (σ ) of W (ǫ) can be defined by
T d σ if σ ≥ 0 ♯ T W (σ ) = sup {ǫ σ − W (ǫ)} = (9) +∞ otherwise, m ǫ ∈R

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By the definition introduced in [10], a function V : Ea → Rm is called canonical if it is Gâteaux differentiable on Ea such that the duality mapping   ∂ Vα (ξ ) β ς = {ςα } = ∇V (ξ ) = : Ea → Ea∗ ⊂ R pα ×m , (13) ∂ξβα is invertible. We note that the geometric variable ξ = {ξβα } is an m × pα matrix, while its β

canonical dual variable ς = {ςα } is a pα ×m matrix. In finite deformation theory and theoretical physics, these are called second order two-point tensors [10]. For the constrained problem (P ) considered in this paper, the dimension pα of the geometrical variable ξ = {ξβα } depends on each given constraint gα (x) ≤ dα , α = 1, . . . , m. Let Iα = {β| β ∈ {1, . . . , pα }} be an index set and let ⎫ ⎧ ⎬ ⎨

α β : Ea × Ea∗ → Rm ξβ ςα ξ ; ς  = ⎭ ⎩ β∈Iα

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denote the partial bilinear form on the product space Ea × Ea∗ . Thus, the Legendre conjugate V ∗ : Ea∗ → Rm of V can be defined by V ∗ (ς) = sta{ξ ; ς  − V (ξ ) : ξ ∈ Ea },

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where the notation sta{∗} denotes computing the stationary points of {∗}. By the assumption that the duality relation (13) is invertible (i.e., canonical), the Legendre conjugate V ∗ (ς ) is uniquely defined on Ea∗ and the inverse duality relation can be written as   ∂ Vα∗ (ς ) ∗ ξ = ∇V (ς ) = . (14) β ∂ςα

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ς = ∇V (ξ ) ⇔ ξ = ∇V ∗ (ς ) ⇔ ξ ; ς  = V (ξ ) + V ∗ (ς ).

In the terminology used in [10], (ξ , ς ) forms a canonical duality pair on Ea × Ea∗ . Thus, noting (12) and (15) to replace g(x) in (10) by V ((x)) = (x); ς  − V ∗ (ς),

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we define the so-called generalized total complementary function (or the extended Lagrang∗ ian [10,27])  : Xa × Rm + × Ea → R as

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(15)

1 (x, σ , ς) = σ T [(x); ς  − V ∗ (ς ) − d] + x T Ax − x T f, (16) 2 m where σ ∈ Rm + is the dual variable vector associated with g(x) ≤ d ∈ R . Through this total complementary function, the canonical dual function can be defined by P d (σ , ς) = sta {(x, σ , ς) : x ∈ Xa } = U  (σ , ς ) − σ T (V ∗ (ς) + d),

(17)

where U  (σ , ς ) is the parametric -conjugate function of the quadratic function U (x) = − 21 x T Ax + x T f defined by the following -conjugate transformation [10,12]:

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It is easy to verify that the following equivalent relations hold on Ea × Ea∗ :

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U  (σ , ς ) = sta{σ T (x); ς  − U (x) : x ∈ Xa }.

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(18)

Ea∗

× be a canonical dual feasible space on which the canonical dual Letting Sc ⊂ function P d (σ , ς) is well defined, the canonical dual problem can be posed as follows: (P d ) : sta{P d (σ , ς ) : (σ , ς) ∈ Sc }.

(19)

Theorem 1 (Complementary-Dual Principle) The problem (P d ) is canonically dual to the primal problem (P ) in the sense that if (¯x, σ¯ , ς¯ ) is a critical point of (x, σ , ς) over ∗ ¯ is a KKT point of (P d ), and ¯ is a KKT point of (P ), (σ¯ , ς) (x, σ , ς ) ∈ Xa × Rm + × Ea , then x P(¯x) = (¯x, σ¯ , ς¯ ) = P d (σ¯ , ς¯ ).

(20)

Proof If (¯x, σ¯ , ς¯ ) is a critical point of  as hypothesized, then we have the following criticality conditions: ∇x (¯x, ς¯ , σ¯ ) = σ¯ T t (¯x); ς¯  + Ax¯ − f = 0,

(21)

¯ σ¯ ) = (¯x) − ∇V ∗ (ς) ¯ = 0, ∇ς (¯x, ς,

(22)

where t (x) = ∇(x) denotes the Gâteaux derivative of , along with the conditions:   0 ≤ σ¯ ⊥ (¯x); ς¯  − V ∗ (ς¯ ) − d ≤ 0, (23)

where the notation ⊥ represents the complementarity or orthogonality condition. Since (ξ , ς ) is a canonical duality pair on Ea × Ea∗ , the criticality condition (22) is equivalent to ς¯ = ∇ξ V ((¯x)) = ∂ V (ξ (¯x))/∂ξ . Substituting this into (21) and using the chain rule to deduce ∇g(¯x) = t (¯x); ∇ξ V ((¯x)), we have, Ax¯ − f + σ¯ T ∇g(¯x) = ∇x L(¯x, σ¯ ) = 0.

This is the criticality condition of the primal problem (P ). By the Legendre equality (¯x); ¯ − V ∗ (ς¯ ) = V ((¯x)), the condition (23) can be written as ς 0 ≤ σ¯ ⊥ (g(¯x) − d) ≤ 0.

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This shows that x¯ is a KKT point of the primal problem (P ). From the complementarity condition (23), we have, (¯x, σ¯ , ς¯ ) = P(¯x).

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On the other hand, by the definition of the canonical dual function, if (¯x, σ¯ , ς¯ ) is a KKT point as hypothesized, the criticality condition (21) leads to ¯ = σ¯ T (¯x); ς¯  − U (¯x). U  (σ¯ , ς)

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Therefore, (¯x, σ¯ , ς¯ ) = P d (σ¯ , ς¯ ) and (σ¯ , ς¯ ) is a KKT point of the dual problem (P d ). ⊓ ⊔

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This theorem shows that there is no duality gap between the primal problem and its canonical dual. In order to identify the global minimizer, we need to study the convexity of the generalized complementary function (x, σ , ς ). Without much loss of generality, we introduce the following assumptions: (i) the geometrical operator (x) : Rn → Ea is twice Gâteaux differentiable; (ii) the canonical function V : Ea → Rm is convex.

Sc+ = {(σ , ς ) ∈ Sc | G a (x, σ , ς )  0, ∀x ∈ Xa }

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Theorem 2 (Global Optimality Condition) Suppose that the conditions in (24) hold and that ¯ is a global maximizer ¯ ∈ Sc+ , then (σ¯ , ς) (¯x, σ¯ , ς¯ ) is a critical point of (x, σ , ς ). If (σ¯ , ς) d + of P on Sc and x¯ is a global minimizer of P on Xc , i.e., P(¯x) = min P(x) = x∈Xc

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max

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P d (σ , ς ) = P d (σ¯ , ς¯ ).

(26)

Proof Actually, this theorem is a special application of the general theory proposed by Gao and Strang in [27]. By the convexity of V (ξ ), its Legendre conjugate V ∗ : Ea∗ → Rm is also T ∗ ∗ convex. Thus, for any given σ ∈ Rm + , the linear combination σ V (ς ) : Ea → R is convex and the generalized complementary function (x, σ , ς ) is concave in ς . By considering σ ∈ Rm + as a Lagrange multiplier for the inequality constraint in Xc , the complementary function (x, σ , ς) can be viewed as a concave (linear) function of σ ∈ Rm + for any given (x, ς) ∈ Xa × Ea∗ . Therefore, for any given x ∈ Rn , we have
P(x) if x ∈ Xc , maxm max∗ (x, σ , ς ) = maxm L(x, σ ) = ∞ otherwise . σ ∈R + σ ∈R+ ς ∈Ea ∗ Moreover, if (σ , ς ) ∈ Sc+ ⊂ Rm + × Ea , then (x, σ , ς ) is convex in x ∈ Xa and concave in m ς for any given σ ∈ R+ . Therefore, if (¯x, σ¯ , ς¯ ) ∈ Xa × Sc+ is a critical point of , we have

(¯x, σ¯ , ς¯ ) = min

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be a subset of Sc . We have the following theorem.

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(24)

By this assumption, we know that the conjugate function V ∗ (ς ) : Ea∗ → Rm is also convex, and for any given (σ , ς ) ∈ Sc , the generalized complementary function (x, σ , ς) is twice Gâteaux differentiable on x. Let G a (x, σ , ς ) = ∇x2 (x, σ , ς) denote the Hessian matrix of (x, σ , ς ) and let

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(x, σ , ς ) = min P(x) x∈Xc

min (x, σ , ς ) =

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P d (σ , ς ).

By Theorem 1, we then have (26).

⊔ ⊓

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(x, σ , ς) =

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1 T x G a (σ , ς )x − σ T (V ∗ (ς) + d) − x T F(σ , ς), 2

where G a (σ , ς ) = A +

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σα Bαβ ςαβ

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(28)

(29)

is the Hessian matrix of (x, σ , ς ), which does not depend on x, and m



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This theorem provides a sufficient condition for a global minimizer of the nonconvex primal problem. In many applications, the geometrical mapping (x) : Xa → Ea is usually a quadratic operator
 1 T α T α (x) = x Bβ x + x Cβ : Rn → Ea ⊂ Rm× pα , (27) 2       α ∈ Rn , and the range Ea depends on where Bαβ = Biαjβ = B αjiβ ∈ Rn×n , Cαβ = Ciβ α α both Bβ and Cβ . In this case, the generalized complementary function has the form:

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F(σ , ς ) = f −

σα Cαβ ςαβ .

(30)

α=1 β∈Iα

The criticality condition (21) in this case is a linear equation of x, i.e., G a (σ , ς )x = F(σ , ς ),

(31)

which is also called the canonical equilibrium equation [10]. Clearly, for a given (σ , ς), if F(σ , ς ) is in the column space of G a (σ , ς), denoted by Col (G a ), the solution of the equation (31) can be written in the form: x = G a+ (σ , ς )F(σ , ς ),

(32)

where G a+ is the Moore–Penrose generalized inverse of G a . Thus, the canonical dual feasible space Sc can be defined as ∗ Sc = {(σ , ς) ∈ Rm + × Ea | F(σ , ς ) ∈ Col (G a )},

(33)

and the canonical dual function P d can be formulated as

1 P d (σ , ς ) = − F(σ , ς)T G a+ (σ , ς )F(σ , ς ) − σ T (V ∗ (ς) + d). 2

(34)

Since (x) is a quadratic operator, its Gâteaux derivative is an affine operator t (x) = ∇(x) = x T Bαβ + CαT β .

By Gao and Strang [27], the complementary operator c (x) of t is defined by 1 c (x) = (x) − t (x)x = − x T Bαβ x. 2

(35)

Thus, the complementary gap function [27] can be defined as

1 1 G(x, σ , ς ) = σ T −c (x); ς  + x T Ax = x T G a (σ , ς )x. 2 2

(36)

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This gap function plays an important role in nonconvex analysis and global optimization. Clearly, G(x, σ , ς ) ≥ 0 ∀x ∈ Xa if G a (σ , ς)  0. Let Sc+ = {(σ , ς ) ∈ Sc | G a (σ , ς )  0},

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= {(σ , ς ) ∈ Sc | G a (σ , ς ) ≺ 0}.

Theorem 3 (Triality Theory) Suppose that (x) is a quadratic operator defined by (27) and the condition (ii) in (24) holds. ¯ ∈ Sc is a critical point of (P d ), then x¯ = G a+ (σ¯ , ς)F( ¯ ¯ is a KKT point of If (σ¯ , ς) σ¯ , ς) ¯ (P ) and P(¯x) = P d (σ¯ , ς). ¯ ∈ Sc+ , then (σ¯ , ς¯ ) is a global maximizer of P d (σ , ς ) on Sc+ , If the critical point (σ¯ , ς) the vector x¯ is a global minimizer of P(x) on Xc , and P(¯x) = min P(x) = x∈Xc

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(39)

P d (σ , ς ) = P d (σ¯ , ς¯ )

(40)

max

¯ P d (σ , ς) = P d (σ¯ , ς).

(41)

x∈Xo

(σ ,ς )∈So

P(¯x) = max P(x) =

(σ ,ς )∈So

¯ ∈ Sc is a critical point of (P d ), we have Proof If (σ¯ , ς)   ¯ = 0, δς P d (σ¯ , ς¯ ) = σ¯ T (¯x) − ∇V ∗ (ς) d

∗

0 ≤ σ¯ ⊥ δσ P (σ¯ , ς¯ ) = (¯x); ς¯  − V (ς¯ ) − d ≤ 0,

(42) (43)

where x¯ = G a+ (σ¯ , ς¯ )F(σ¯ , ς¯ ). Equation (42) asserts that if σ¯ α  = 0, then the corresponding ¯ is a canonical duality pair on Ea × Ea∗ , ξα (¯x) = α (¯x) = ∇ςα V ∗ (ς¯ ). By the fact that ((¯x), ς) ¯ = V ((¯x)) = g(¯x). Therefrom the equivalent relations in (15), we have (¯x); ς¯ − V ∗ (ς) fore, the complementarity condition in (43) leads to σ¯ α (gα (¯x) − dα ) = 0. If σ¯ α  = 0, we must have the criticality condition gα (¯x) − dα = 0. This shows that if (σ¯ , ς¯ ) is a critical point of P d (σ , ς), the vector x¯ = G a+ (σ¯ , ς¯ )F(σ¯ , ς¯ ) is a KKT point of (P ). Since the total complementary function (x, σ , ς ) is a saddle function on Xa × Sc+ and a super-critical function on Xa × Sc− (i.e.,  is concave in both x ∈ Xa and (σ , ς ) ∈ Sc− ) , the remainder of the theorem follows from the triality theory developed in [10,12,16]. ⊔ ⊓ Remark 1 Note that the dual variable σ ∈ Rm + , i.e., it is restricted by the inequality constraint ¯ is σ ≥ 0 ∈ Rm . From the proof of Theorem 1 (as well as Theorem 3) we know that if (¯x, σ¯ , ς) a critical point of , then x¯ and ς¯ satisfy the criticality conditions (21) and (22), respectively, while σ¯ satisfies the KKT conditions (23). Given a nonconvex quadratic objective function P(x), the global minimizer x¯ is located on the boundary of the feasible set Xc , i.e., there exists at least one σ¯ α > 0 such that ¯ − dα = gα (¯x) − dα = 0. ∇σα (¯x, σ¯ , ς¯ ) = α (¯x); ς¯α  − Vα∗ (ς)

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P d (σ , ς ) = P d (σ¯ , ς¯ ).

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¯ ∈ Sc− , then on the neighborhood Xo × So of (¯x, σ¯ ), we have If the critical point (σ¯ , ς) either

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Then from Theorems 1, 2, and the triality theory developed in [10,12], we have the following result:

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i.e., the dual variable σ¯ α is a critical point of (x, σ , ς ), and the corresponding vector x¯ is KKT point of P(x). By the fact that the canonical dual feasible space Sc+ is a closed convex

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Sc‡ = {(σ , ς ) ∈ Sc | G a (σ , ς) ≻ 0},

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the Hessian matrix G a (σ , ς ) is invertible and the canonical dual function can be formulated as 1 P d (σ , ς) = − F(σ , ς )T G a−1 (σ , ς )F(σ , ς ) − σ T (V ∗ (ς ) + d). 2

(45)

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¯ could be located on the boundary of Sc+ . In this case, the matrix set, the critical point (σ¯ , ς) G a (σ , ς ) may be singular and the canonical dual problem may have more than one critical point on Sc+ . Particularly, on the open canonical dual feasible space

In this case, the canonical dual problem   d (P+ ) : max P d (σ , ς ) : (σ , ς ) ∈ Sc‡

(46)

usually has at most one solution (σ¯ , ς¯ ), which is a critical point of P d (σ , ς ). By Theorem 3, the vector x¯ = G a−1 (σ¯ , ς¯ )F(σ¯ , ς¯ )

is a solution to (P ), which is located on the boundary of Xc under the nonconvexity of P(x). The related existence conditions will be discussed in the following section. The generalized complementary function (x, σ , ς ) defined in (16) is actually the secondorder Lagrangian in the canonical duality theory (see Sect. 4.1.2 in [10]). For the quadratic geometrical mapping (x), the canonical dual function P d (σ , ς ) can be formulated explicitly, which is also called the pure complementary energy function in finite deformation theory (see [7–9]). The analytical solution of type (32) was first obtained in nonconvex variational problems [7,11] and finite deformation mechanics [9]. Similar results also appear in global optimization [12,16,17]. It is interesting to note that many nonconvex primal problems in nonconvex analysis and global optimization share the same form of the canonical dual function as defined in (34) (see [8,12,20,26]). In the case that (x) is a higher order nonlinear function, the sequential canonical dual transformation can be used to formulate a higher order
complementary function (see Chap. 4 in [10]).

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Theorem 3 shows that by the canonical dual transformation, the nonconvex primal problem (P ) can be transformed to a concave maximization dual problem over a convex feasible space Sc+ , which can be solved via well-developed nonlinear programming techniques (cf. [40]).

302

3 Applications to quadratic constrained problems

298 299 300

303 304

305

306 307 308

We begin by considering the following nonconvex quadratic minimization problem with quadratic inequality constraints, denoted by (Pq ):  
min P(x) = 21 x T Ax − x T f (47) (Pq ) : s.t. 21 x T Bα x + x T Cα ≤ dα , α = 1, . . . , m,       where Bα = Biαj = B αji ∈ Rn×n , Cα = Ciα ∈ Rn , ∀ α = 1, . . . , m, and d = {dα } ∈ Rm is a vector. Due to the nonconvex cost function and nonconvex inequality constraints, this problem is known to be NP-hard [42].

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Compared with (27), we have pα = 1 and the canonical function V (ξ ) = ξ is a self-mapping. Therefore, the canonical dual variable ς = ∇V (ξ ) = I is an identity matrix in Rm×m and V ∗ (ς) = sta{ξ ; ς  − ξ | ξ ∈ Rm } = 0. In this case, the generalized complementary function (28) has a very simple form: q (x, σ ) =

316

317

where G q (σ ) = A +

318

319

320

321

330 331 332 333 334

G(x, σ ) =

337

σα Cα .

(50)

α=1

(51)

(52)

1 T x G q (σ )x, 2

(53)

which is nonnegative on Rn if G q (σ )  0. Let

Sq+ = {σ ∈ Sq | G q (σ )  0}, and Sq− = {σ ∈ Sq | G q (σ ) ≺ 0}.

(54)

Then the canonical dual problem for this quadratic constrained problem is given by (Pqd ) : max{Pqd (σ ) : σ ∈ Sq+ }.

(55)

From Theorem 3 we have the following result:

Theorem 4 The problem (Pqd ) is canonically dual to the primal problem (Pq ) in the sense that for each critical point σ¯ ∈ Sq of (Pqd ), the vector x¯ = G q+ (σ¯ )Fq (σ¯ ) is a KKT point of (Pq ) and P(¯x) = P d (σ¯ ). Particularly, if the critical point σ¯ ∈ Sq+ , then σ¯ is a global maximizer of (Pqd ), the vector x¯ is a global minimizer of (Pq ), and P(¯x) = min P(x) = max Pqd (σ ) = Pqd (σ¯ ). x∈Xc σ ∈Sq+

335

336

m

In this case, the complementary gap function has a simple form:

328

329

α=1

and Fq (σ ) = f −

1 Pqd (σ ) = − Fq (σ )T G q+ (σ )Fq (σ ) − σ T d. 2

326

327

σα Bα ,

(49)

the canonical dual function Pqd can be formulated as

324

325

m

Therefore, on the dual feasible space   Sq = σ ∈ Rm + | Fq (σ ) ∈ Col (G q ) ,

322

323

1 T x G q (σ )x − x T Fq (σ ) − σ T d, 2

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Since the constraint g(x) is a vector-valued quadratic function defined on Xa = Rn , we simply let 
1 T α x B x + x T Cα : Rn → Rm . (48) g(x) = (x) = 2

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(56)

If G q (σ¯ ) ≻ 0, then σ¯ is a unique global maximizer of (Pqd ) and the vector x¯ = G q−1 (σ¯ )Fq (σ¯ ) is a unique global minimizer of (Pq ).

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350 351

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P(¯x) = min P(x) = min Pqd (σ ) = Pqd (σ¯ ). x∈Xo σ ∈ So

This theorem shows that the Hessian matrix of the complementary gap function G(x, σ ) provides sufficient and uniqueness conditions for globally minimizing the quadratic constrained problem (Pq ). We note that this theorem is actually a special application of the general canonical duality theory developed in [10,12,16,27]. In order to study the existence theory, we need to introduce the following sets:   ∂ Sq = σ ∈ Rm | det G q (σ ) = 0 , (58)   + ∂ Sq = σ ∈ Sq | det G q (σ ) = 0 . (59) Similar to the general results proposed in [29], we then have the following result:

Theorem 5 Suppose that for the given matrices A, {Bα }, {Cα } and vectors f, d, there exists at least one σ 0 ∈ Sq+ such that G q (σ 0 )  0 and lim

σ  → ∞ σ ∈ Sq+

353 354 355 356

357

359

360 361 362 363 364 365 366 367 368

369 370

371

Pqd (σ ) = −∞.

(60)

Then the canonical dual problem (Pqd ) has at least one KKT point σ¯ ∈ Sq+ . If σ¯ ∈ Sq+ is also a critical point of Pqd (σ ), then x¯ = G q+ (σ¯ )Fq (σ¯ ) is a global minimizer for the primal problem (Pq ). + Moreover, if ∂ Sq ⊂ Rm + , there exists at least one σ 0 ∈ Sq such that G q (σ 0 ) ≻ 0, and lim

σ → ∂ Sq+ σ ∈ Sq+

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(57)




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However, if the critical point σ¯ ∈ Sq− , then σ¯ is a local minimizer of Pqd (σ ) on the neighborhood So ⊂ Sq− if and only if x¯ is a local minimizer of P(x) on the neighborhood Xo ⊂ Xc , i.e.,

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Pqd (σ ) = −∞,

(61)

then the canonical dual problem (Pqd ) has a unique global maximizer σ¯ ∈ Sq+ and x¯ = G q−1 (σ¯ )Fq (σ¯ ) is a unique global minimizer for the primal problem (Pq ). Proof By the fact that the feasible space Sq+ is a semi-closed convex set whose boundary ∂ Sq+ is a hyper-surface in Rm , if there exists a σ 0 such that G q (σ 0 )  0, then Sq+ is not empty. Since the canonical dual function Pqd (σ ) is continuous and concave on Sq+ , which is finite on ∂ Sq+ , if the condition (60) holds, then Pqd (σ ) has at least one maximizer on Sq+ . m + Moreover, if ∂ Sq ⊂ Rm + , then Sq ⊂ R+ . If there exists a σ 0 such that G q (σ 0 ) ≻ 0, + then Sq is non-empty and has at least one interior point. Under the conditions (60) and (61), the canonical dual function Pqd (σ ) is strictly concave and coercive on the open convex set Sq+ \∂ Sq+ . Therefore, the canonical dual problem (Pq ) has a unique maximizer σ¯ ∈ Sq+ that is a critical point of Pqd (σ ). ⊔ ⊓ Theorem 5 shows that under the conditions (60) and (61), the canonical dual function Pqd (σ ) has a unique maximizer σ¯ on the open feasible space Sq‡ = {σ ∈ Sq | G q (σ ) ≻ 0}.

(62)

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Particularly, if m = 1 and C = 0, the problem (Pq ) has only one quadratic constraint g(x) = 21 x T Bx ≤ d. Therefore, the canonical dual function has only one variable [18]: 1 Pqd (σ ) = − f T (A + σ B)−1 f − σ d, 2

378

379

380 381 382 383

384 385 386 387

388

389 390 391 392 393

(64)

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and the criticality condition ∇ Pqd (σ ) = 0 leads to a nonlinear algebraic equation

1 T (65) f (A + σ B)−1 B(A + σ B)−1 f = d, 2 which can be solved easily to obtain all dual solutions. Moreover, if B = I is an identity matrix in Rn , then the constraint 21 x T Bx = 21 x2 ≤ d is an n-dimensional sphere. It was shown in [17,18] that number of solutions depends on the number of negative eigenvalues of A. These canonical dual solutions play an important role in trust region methods.

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In this case, the matrix G q (σ ) is invertible on Sq‡ and the canonical dual function Pqd can be written as 1 Pqd (σ ) = − Fq (σ )T G q−1 (σ )Fq (σ ) − σ T d. (63) 2

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Remark 2 It is instructive to see connections between (63) and the classical Lagrangian Theory proposed in (3)–(5). Under the assumption that det G q (σ )  = 0, the necessary optimality condition ∇x L(x, σ ) = 0 in (3) leads to the unique solution x = G q−1 (σ )Fq (σ ). Substituting this into (4) we get
d Pq (σ ) if σ ∈ Sq‡ , P ∗ (σ ) = inf L(x, σ ) = x∈Xa −∞ otherwise. Therefore, under the sufficient (global) optimality condition σ ∈ Sq‡ , the Lagrangian dual function P ∗ (σ ) is identical to the canonical dual function Pqd (σ ) as defined in (63). By Theorem 4, we know that if σ¯ ∈ Sq‡ is a critical point of Pqd (σ ), then the corresponding

x¯ = G q−1 (σ¯ )Fq (σ¯ ) ∈ Xc satisfies σ¯ ⊥ (g(¯x) − d), and (¯x, σ¯ ) ∈ Rn × Sq‡ is a saddle point solution with the following strong duality condition holding: inf P(x) = sup P ∗ (σ ). σ ∈Sq‡

394

x∈Xc

395 396

However, in many applications if {Bα } (α = 1, . . . , m) are either indefinite or zero matrices, the dual feasible space Sq‡ could be an empty set and in this case, P ∗ (σ ) = inf L(x, σ ) = −∞.

397

x∈Xa

398 399 400 401

Therefore, the classical Lagrangian duality theory cannot be used to solve the primal problem (see Example 2 in the next section). But, by the complementary-dual principle (the first part of Theorem 4) we know that the primal problem is equivalent to the following canonical dual minimal stationary problem: min sta{Pqd (σ ) : σ ∈ Sq }.

402

403 404 405

(66)

For each critical point σ¯ ∈ Sq , the vector x¯ = G q−1 (σ¯ )Fq (σ¯ ) is a KKT point of the primal problem. If σ¯ ∈ Sq− is a local minimizer of Pqd (σ ), by the triality theory we know that x¯ is a local minimizer of (Pq ). ⊔ ⊓

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Fig. 1 Graph of P(x) (left); contours of P(x) and boundary of Xc (right) for Example 1

407 408 409 410 411 412 413 414 415 416 417 418

419

420

421 422 423 424

425

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428 429 430 431

432

This remark shows that for nonconvex minimization problems with quadratic constraints, the Lagrangian dual P ∗ (σ ) is identical to the canonical dual P d (σ ) only on the canonical dual feasible space Sq‡ . The sufficient global optimality conditions in Sq‡ close the duality gap that exists in the classical Lagrangian duality theory. The triality theory (Theorem 4) can be used to develop effective canonical dual approaches for solving a wide class of primal problems. The general sufficient global optimality conditions for nonconvex minimization problems was first proposed by Gao and Strang in [27], where G(x, σ ) is called the complementary gap function associated with the quadratic operator (x). They discovered that under the condition G(x, σ ) ≥ 0, ∀x ∈ Xa , the critical point of the total complementary function q (x, σ ) leads to global optimal solutions to the primal problem. We note that similar results and the so-called L-subdifferential condition proposed in the recent papers [34,35] are actually special applications of the general canonical duality theory developed in [10,27].

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Example 1 In 2-D space, if we let       3 0.5 1 0 1 A= , B= , and f = , 0.5 −2.0 0 0.5 1.5 then the matrix A is indefinite, while B is positive definite. Setting d = 2, the graph of the primal function P(x) = 21 x T Ax − x T f is a saddle surface (see Fig. 1), and the boundary of the feasible set Xc = {x ∈ R2 | 21 x T Bx ≤ d} is an ellipse (see Fig. 1). In this case, the canonical dual function (64) can be formulated as  −1   1 3 + σ 0.5 1 Pqd (σ ) = − (1 1.5) − 2σ, 0.5 −2 + 0.5σ 1.5 2 which has four critical points (see Fig. 2):

σ¯ 1 = 5.08 > σ¯ 2 = 3.06 > σ¯ 3 = −2.46 > σ¯ 4 = −3.68.

Since G q (σ¯ 1 ) ≻ 0 and G q (σ¯ 4 ) ≺ 0, the triality theory yields that x1 = (−0.05, 2.83)T is a global minimizer located on the boundary of Xc , and x4 = (−1.95, −0.64)T is a local maximizer. From the graph of P d (σ ) we can see that x2 = (0.39, −2.77)T is a local minimizer, and x3 = (2.0, −0.16)T is a local maximizer. We have P(x1 ) = −12.25 < P(x2 ) = −4.24 < P(x3 ) = 4.04 < P(x4 ) = 8.81.

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Fig. 2 Graph of P d (σ )

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438 439 440 441

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445 446 447

448

449 450 451 452

453

4 Applications to box and integer constrained problems

We now turn our attention to the box constrained quadratic minimization problem:
  min P(x) = 21 x T Ax − x T f (Pb ) : s.t. − 1 ≤ xi ≤ 1, i = 1, . . . , n.

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An associated problem is the quadratic integer (Boolean) programming problem
  min P(x) = 21 x T Ax − x T f (Pi ) : s.t. xi ∈ {−1, 1}, i = 1, . . . , n.

(67)

(68)

Both problems are known to be NP-hard and have been subjected to considerable study over the past several decades (see [47,48]). In order to use the canonical dual transformation, the key step is to rewrite the linear box constraints in the following canonical form:
 1 T α g(x) = (x) = x B x ≤ d, (69) 2 where Bα = {Biαj } ∈ Rn×n , ∀ α = 1, . . . , n, i.e.,
1 if i = j = α, α Bi j = 0 otherwise,  n and d = 21 is a vector whose components are all 21 . Thus, the box constrained problem is actually the most simple case of the problem (Pq ). Particularly, if the inequality constraint in (69) is replaced by
 1 T α g(x) = (x) = (70) x B x = d, 2

then x has to be in {−1, 1}n . Therefore, the Boolean programming problem (Pi ) is actually a special case of the quadratically constrained quadratic program (where each equality constraint is equivalently represented by two inequalities). The canonical dual problem in this case has an even simpler form (see [20]):
 1 T (71) (Pbd ) : max Pbd (σ ) = − f T G + (σ )f − σ d : σ ∈ S b , b 2

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G b (σ ) = A +

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458 459 460 461 462 463

464 465 466

467

468

469

470 471 472 473

474 475

476

477

478

479 480 481 482 483 484 485 486 487

σα Bα ,

(72)

α

Sb = {σ ∈ Rn+ | f ∈ Col (G b )}.

(73)

Theorem 6 If σ¯ ∈ Sb is a critical point of the canonical dual problem (Pbd ), then x¯ = G+ x) = Pbd (σ¯ ). If σ¯ > 0, b (σ¯ )f is a KKT point of the box constrained problem (Pb ) and P(¯ n then x¯ ∈ {−1, 1} is a KKT point of the Boolean programming problem (Pi ).

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If G b (σ¯ ) ≻ 0, then x¯ = G −1 b (σ¯ )f is a global minimizer of the problem (Pb ). If G b (σ¯ ) ≻ 0 and σ¯ > 0, then x¯ = G −1 b (σ¯ )f is a global minimizer of the problem (Pi ). ⊔ ⊓ The proof of this theorem can be found in [20]. This theorem shows that by the canonical dual transformation, the nonconvex integer programming problem (Pi ) can be converted to the following continuous concave maximization dual problem:
 1 T −1 + d d T (Pi ) : max Pb (σ ) = − f G b (σ )f − σ d : σ ∈ Si , (74) 2

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where

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where Si+ = {σ ∈ Rn+ | σ > 0, G b (σ ) ≻ 0}. By the fact that d = { 21 }n > 0 and lim

σ →∞

Pbd (σ ) = −∞,

we know from Theorem 5 that the canonical dual problem (Pbd ) has at least one KKT point σ¯ ∈ Sb+ . If σ¯ ∈ Si+ ⊂ Sb+ , then x¯ = G −1 b (σ¯ )f is a global minimizer for (Pi ). A detailed study on canonical dual approaches for 0–1 programming and multi-integer programming is given in [3,25,49]. Example 2 For a given vector f ∈ Rn , we consider the following constrained convex maximization problem: max{x + f2 : x∞ ≤ 1},

(75)

which is equivalent to the following concave quadratic minimization problem:
 1 T T T min P(x) = − x x − x f : |xi | ≤ 1, ∀i = 1, . . . , n, x x ≤ r , 2

(76)

where r > n to ensure that the additional quadratic constraint x T x ≤ r in the feasible space ∀i = 1, . . . , n, x T x ≤ r } is never active. Therefore, (76) is indeed a box constrained concave minimization problem. It is known that for high dimensional nonconvex constrained optimization problems, to check which constraints are active is fundamentally difficult [42]. If we let n = 2, r = 100, and f = (1, 1)T , the optimal solution is x¯ = (1, 1)T with objective value P(¯x) = −3. To illustrate the difficulty of applying the classical Lagrangian duality theory directly to (76), we first introduce Lagrange multipliers (σ1 , σ2 , σ3 , σ4 )T ∈ R4+ to relax the linear box constraints −1 ≤ xi ≤ 1, i = 1, 2, and σ5 ≥ 0 to relax the quadratic Xc = {x ∈ Rn | − 1 ≤ xi ≤ 1,

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2

  1 σi (xi − 1) − σi+2 (xi + 1) L(x, σ ) = − (x12 + x22 ) − (x1 + x2 ) + 2

490

with the Lagrangian dual function given by

P ∗ (σ ) = min L(x, σ ).

492

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x∈R2

493

When σ5 < 1, we get P ∗ (σ ) = −∞. When σ5 = 1, we obtain max{P ∗ (σ ) : σ5 = 1} = −52 σ ≥0

494

495 496

of

i=1

 1  + σ5 x12 + x22 − 100 , 2

at the solution σ0 = (1, 1, 0, 0, 1). Finally, for any given σ ∈ Sr = {σ ∈ R5+ | σ5 > 1}, the Lagrangian dual function can be obtained as 4

497

498

P ∗ (σ ) = −

501 502 503

sup{P ∗ (σ ) : σ ∈ Sr } = −52,

realized as σ → σ o = (1, 1, 0, 0, 1)T . Since σ o is not a critical point of P ∗ (σ ), both Theorems 4 and 6 do not apply for this case and the associated xo is not a KKT point of the primal problem (76). Hence, there exists a duality gap between the primal and the Lagrangian dual problem, i.e., P(¯x) = min P(x) = −3 > −52 = max P ∗ (σ ) = P ∗ (σ o ). x∈Xc σ ∈R5+

504

505

506

507

508

i=1

It is easy to check that the solution to the Lagrangian dual problem

499

500

 

1 σi − 50σ5 . (1 − σ1 + σ3 )2 + (1 − σ2 + σ4 )2 − 2(σ5 − 1)

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constraint 21 x T x ≤ 50. The Lagrangian associated with (76) is

To close this duality gap, we rewrite the constraints in the canonical form ⎫ ⎧1 ⎫ ⎧1 2 ⎪ ⎪ ⎨2 ⎪ ⎬ ⎪ ⎨ 2 x1 ⎬ ≤ 21 g(x) = (x) = 21 x22 =d ⎪ ⎪ ⎩ ⎪ ⎭ ⎪ ⎩1 2 ⎭ 2 50 2 (x 1 + x 2 )

as defined in (69) with
1 if i = j = α, Biαj = 0 otherwise,

i, j, α = 1, 2,

Bi3j =




1 if i = j, 0 if i  = j,

i, j = 1, 2

511

and d = 0.5(1, 1, 100)T . In effect, this reformulates (76) for this instance as follows:
 1 2 1 1 2 1 2 1 2 1 2 2 x ≤ , (x + x2 ) ≤ 50 . min P(x) = − (x1 + x2 ) − (x1 + x2 ) : x1 ≤ , 2 2 2 2 2 2 2 1 (77)

512

The associated total complementary function (49) has a simple form:

509

510

513

q (x, σ ) =

1 1 1 2 x1 (σ1 + σ3 − 1) + x22 (σ2 + σ3 − 1) − (x1 + x2 ) − (σ1 + σ2 ) − 50σ3 . 2 2 2

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The optimal solution for this concave maximization problem is σ¯ = (2, 2, 0)T with the optimal value Pqd (σ¯ ) = −3. Observe that σ¯ 3 = 0 reflects the fact that the quadratic constraint x T x ≤ r is inactive. Since σ¯ ∈ Sq+ is a critical point of Pqd (σ ), therefore, the vector x¯ = G q−1 (σ¯ )f = (1, 1)T is a global minimizer of the primal problem with zero duality gap. Finally, it is insightful to note that once the primal problem has been reformulated as (77), the classical Lagrangian dual is then given by max{P ∗ (σ ) : σ ≥ 0}, where P ∗ (σ ) = minx {q (x, σ )}. It is readily verified that for σ ∈ Sq+ , P ∗ (σ ) is given by Pqd (σ ) as in (78), while P ∗ (σ ) = −∞ otherwise. Hence, under the sufficient global optimality condition σ ∈ Sq+ , the Lagrangian dual for the canonically reformulated problem (77) is precisely given by the canonical dual (78).

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Then, on the canonical dual feasible space Sq+ = {σ ∈ R3+ | σi + σ3 − 1 > 0, i = 1, 2} the canonical dual problem (Pqd ) is   
1 1 1 1 − (σ1 + σ2 ) − 50σ3 : σ ∈ Sq+ . max Pqd (σ ) = − + 2 σ1 + σ3 − 1 σ2 + σ3 − 1 2 (78)

Remark 3 This example shows the difficulty of directly applying the classical Lagrangian duality for solving nonconvex minimization problems with linear (including both box and integer) constraints. The classical Lagrangian duality theory was originally developed for linearly constrained convex variational problems in analytical mechanics, where the Lagrange multipliers and the linear constraints possess certain perfect duality (work-conjugate) relations [10,38]. In convex analysis, the classical (saddle) Lagrangian duality theory was generalized for solving the following convex minimization problem [2,44]:

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min{P(x) = W (x) − F(x)},

where  is a linear operator, W (y) is a convex (nonsmooth) function, and F(x) is a concave (or linear) function. By using the Fenchel transformation, the so-called Fenchel– Moreau–Rockafellar dual problem is formulated as max{P d (σ ) = F ♯ (∗ σ ) − W ♯ (σ )},

where ∗ is an adjoint operator of . In constrained problems, if the function W (x) is an indicator of the linear (geometrical) constraint x ≤ d as defined by Equation (7), then by the Lagrange multiplier law (see [10], p. 36–37), the Lagrange multiplier σ for the primal problem should be a solution for the dual problem. Dually, if the Fenchel conjugate F ♯ is an indicator of the balance constraint ∗ σ = f [10], then the Lagrange multiplier x for the dual problem should be a solution to the primal problem. The convexity of the primal function P(x) leads to the strong duality min P(x) = max P d (σ ). In physics and continuum mechanics, the perfect dual function is called the complementary energy and the strong duality relation is controlled by certain conservation laws. However, the Lagrange multiplier method has been misused for solving general nonlinear constrained nonconvex problems. The primal problem (76) in Example 2 has both linear and nonlinear (quadratic) constraints, the Lagrange multipliers σi , i = 1, 2, 3, 4 are dual to the linear constraints, while σ5 is dual to the quadratic constraint. Since the linear and nonlinear constraints are different geometrical measures, their corresponding dual variables, i.e., the Lagrange multipliers σi , i = 1, 2, 3, 4, and σ5 are in different metric spaces with different (physical) units. Therefore, the classical Lagrangian dual problem in this case does not make physical sense. The weak Lagrangian duality theory leads to various duality gaps and violates the conservation law.

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Generally speaking, if the objective function P(x) is nonconvex, the optimal solutions are usually located on the boundary of the feasible set. To determine active constraints is a fundamentally difficult task. For a nonconvex polynomial P(x), the effective domain of the Lagrangian dual P ∗ (σ ) = supσ L(x, σ ) is usually empty. Therefore, many extended Lagrangian duality methods have been developed. As indicated in [12,16,27], the key step in the canonical dual transformation is to choose a (nonlinear) geometrical operator ξ = (x) such that the duality relation ς = ς (ξ ) defined by (13) is invertible (one-to-one) and the nonconvex primal problem can be written in the canonical form (12). Detailed discussion on the geometrical operator (x) and the canonical dual pairs (ξ , ς ) are given in [10] (Chap. 6). For most of problems in nonconvex analysis and global optimization, the quadratic geometrical operator (x) can be used to formulate canonical dual problems [12,27]. By the fact that the total complementary function is identical to the Lagrangian for the reformulated primal problem (77), the function q (x, σ ) was also called the nonlinear (or extended) Lagrangian in [10]. In physics and continuum mechanics, since the terminology complementarity represents perfect duality, it is more appropriate to name q (x, σ ) as the total complementary function. According to the canonical duality theory, when the primal problem is reformulated in the canonical form, the canonical dual function and its feasible space can be formulated precisely via the canonical dual transformation. Both global and local extremality conditions are governed by the triality theory. It is interesting to note that by using the quadratic geometrical measure, different primal problems in differential fields can be transformed to a unified canonical dual form. Actually, the canonical dual function Pqd (σ ) was first proposed in nonconvex analysis and mechanics [7–9], which leads to analytical solutions for a class of nonconvex variational/boundary value problems [11,22,23]. In continuum mechanics, the quadratic measure ξ = (x) corresponds to the CauchyRiemann type strain tensor, while its canonical dual ς = ∇V (ξ ) is the well-known second Piola–Kirchhoff stress tensor. For most of hyper-elastic materials, the strain energy V (ξ ) is a convex function of the canonical strain measure ξ (see [10], Chap. 6). In plasticity of solids, the box constraint x√ ∞ ≤ 1 corresponds to the well-known Tresca yield condition, while the condition x2 ≤ n corresponds to the von Mises yield condition (see [10], p. 404). In the famous experiment by Taylor and Quinney (1931), the experimental data for most metals lies between the two criteria; however, the data are generally closer to the von Mises yield condition. By using the quadratic operator (x) = x22 , the canonical dual transformation was first proposed to solve nonlinear finite element programming problems in large scale plastic limit analyses of structures [5,6]. The vector-valued quadratic canonical dual transformation (x) = {xi2 } for solving box and integer constrained problems was presented in [3,20,25,49]. ⊔ ⊓

pro

558

unc orre cted

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5 Nonconvex polynomial constrained problems

We now assume that g(x) is a general fourth order polynomial constraint given by ⎫ ⎧ 2 ⎬  ⎨ 1 1 T α Dβ x Bβ x + x T Cαβ − E βα g(x) = ≤ d, ⎭ ⎩ 2 α 2

(79)

β∈Iα

597 598

α } ∈ Rn are given as before, I is a (finite) index where Bαβ = {Biαjβ } ∈ Rn×n and Cαβ = {Ciβ α β

set that depends on each index α = 1, . . . , m, and {Dα } and {E βα } are two given second order

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603 604

605

606

607

where the range of Ea depends on the tensors {Bαβ } and {Cαβ }, the canonical function ⎫ ⎧ ⎨ 1 2 ⎬  D β ξ α − E βα : Ea → Rm V (ξ ) = ⎭ ⎩ 2 α β β∈Iα

608

609

610 611

612

is a quadratic function. Thus, the canonical duality relation

ς = ∇V (ξ ) = {Dαβ (ξβα − E βα )} : Ea → Ea∗

β

is a linear mapping, where the range of Ea∗ depends on the tensors {Dα } and {E βα }. The Legendre conjugate V ∗ can be defined uniquely as: ⎧ ⎫  ⎬ ⎨

1 α β β 2 ) + E ς (ς . (80) V ∗ (ς ) = α α β β ⎭ ⎩ 2Dα

unc orre cted

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602

of

600 601

β

tensors. We assume that Dα > 0, ∀α ∈ {1, . . . , m}, β ∈ Iα . This function is also called the fourth order canonical polynomial as discussed in [10,19], which arises in many applications (e.g., in Euclidean distance problems [28,36,45], post-buckling analysis in large deformation beam theory [13], finite element analysis for phase transitions of super-conductivity [29], and sensor network localization [30]). By introducing a geometrical measure
 1 T α ξ = (x) = {ξβα } = x Bβ x + x T Cαβ : Rn → Ea , 2

pro

599

β∈Iα

613

614

Substituting this into (34), the canonical dual function has the following form:    m

1 1 β 2 α β d T + (ς ) +E β ςα +dα P (σ , ς )=− F(σ , ς ) G a (σ , ς )F(σ , ς )− , σα β α 2 2Dα α=1 β∈Iα

615

616

617

618 619 620

621

622 623 624 625 626

627

(81)

which is concave on the dual feasible space

∗ Sc+ = {(σ , ς ) ∈ Rm + × Ea | F(σ , ς) ∈ Col (G a ), G a (σ , ς )  0}.

(82)

Remark 4 Similar to Remark 2, it is again insightful to view the connection between the canonical dual (81) and the classical Lagrangian dual defined by (3)–(5). In this case, we have as in (3) that L(x, σ ) =

1 T x Ax − x T f + σ T (g(x) − d) . 2

(83)

Clearly, without introducing the canonical dual pair (ξ , ς), the Fenchel-Moreau-Rockafellar dual P ∗ (σ ) = inf x∈Xa L(x, σ ) defined by (4) cannot be defined explicitly due to the high order nonlinearity of the constraint g(x) ≤ d. However, by using the canonical dual transformation ξ = (x) and the chain rule, the necessary condition for the optimization in (4), i.e., ∇x L(x, σ ) = 0 leads to Ax − f +

m  

σα ςαβ (Bαβ x + Cαβ ) = 0.

(84)

α=1 β∈Iα

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This is the canonical equilibrium equation G a (σ , ς )x = F(σ , ς ), where ς≡

630 631

∂ V ((x)) ∂ξ

(85)

is as defined above. If G a (σ , ς) is invertible for (σ , ς ) ∈ Sc as defined in (33), we get from (84) that x uniquely satisfies x = G a−1 (σ , ς )F(σ , ς ). Furthermore, by (84), we get m 1 1

1 T σ α ς βα [x T Bβα x + x T Cβα ]. x Ax = x T f − 2 2 2

632

633 634 635 636 637 638 639 640 641 642 643 644

645

646

647

648

649 650 651

Substituting for 21 x T Ax in (83) using (86), and then applying the optimality condition x = G a−1 (σ , ς)F(σ , ς), the Lagrangian dual function reduces precisely to P d (σ , ς ) defined in (81) under the global optimality condition (σ , ς ) ∈ Sc+ . Note that ∇ς (x, σ , ς ) = 0 produces the inverse of the identity (85) under the relevant case when σ α  = 0, ∀α = 1, . . . , m, thus validating the foregoing derivation. In finite deformation theory, the geometrical measure ξ = (x) is called the canonical strain tensor and the canonical duality relation (85) is called the constitutive law. This one-to-one constitutive relation leads to the canonical dual function P d . The associated Theorem 1 is called the pure complementary variational principle, first developed in nonconvex variational analysis [7,8]. This theorem solves an open problem in nonconvex mechanics posed by Hellinger (1914) and Reissner (1953) (see [43]) and it is known as the Gao principle [39]. ⊔ ⊓ We now present some special case applications for (79).

5.1 Quadratic minimization with one nonconvex polynomial constraint We first assume that the primal problem has only one nonconvex constraint:  2 1 1 T T g(x) = x Bx + x c − η ≤ d, 2 2

G a (σ, ς) = A + σ ς B, F(σ, ς) = f − σ ςc.

The canonical dual function is

1 P d (σ, ς) = − FT (σ, ς)G a+ (σ, ς)F(σ, ς) − σ 2

654

655 656 657 658



 1 2 ς + ης + d . 2

(88)

Example 3 In 2-D space, let B be an identity matrix, c = 0, and let A be a diagonal matrix with a11 = 0.6, a12 = a21 = 0, and a22 = −0.5. Setting f = (0.2, −0.1)T , d = 1, and η = 1.5, the constraint g(x) ≤ d is an annulus (see Fig. 3 (right)). Solving the dual problem, we get σ¯ = 0.3829201, and ς¯ = 1.4142136.

659

660

(87)

where B is an n × n matrix, c ∈ Rn is a vector, and η > 0 is a constant. In physics, this nonconvex fourth order polynomial is known as the double-well function and it appears in many applications [10]. In this case, m = |Iα | = 1, and

652

653

(86)

pro

α=1 β∈Iα

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The primal solution

x¯ =

661



0.1752033 −2.4078477



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Fig. 3 Graph of P(x) (left); contours of P(x) and boundary of Xc (right) for Example 3

662

is located on the boundary of the feasible set Xc (see Fig. 3) and we have P(¯x) = −1.7160493 = P d (σ¯ , ς¯ ).

663

5.2 Combined quadratic and nonconvex polynomial constraints

665

We now consider the problem with the following two constraints:

1 T 1 x B x + x T C1 ≤ d1 , 2 2  1 1 T 2 T 2 g2 (x) = x B x + x C − η ≤ d2 . 2 2 g1 (x) =

666

667

668

669

670

671

672 673

In this case, m = 2, Iα = {1}, for α = 1, 2, and the geometrical operator
 1 α T α ξ = (x) = xB x + x C : Rn → R2 2 is a 2-vector. The canonical function V (ξ ) is a vector-valued function 
1 V (ξ ) = ξ1 , (ξ2 − η)2 . 2 The canonical dual variable is ς = ∇V (ξ ) = {1, ξ2 − η}. Since ς1 = 1, we let ς2 = ς. Thus, the canonical dual function has only three variables (σ1 , σ2 , ς) ∈ R3 , i.e., 1 P d (σ1 , σ2 , ς) = − F(σ1 , σ2 , ς)T G a+ (σ1 , σ2 , ς)F(σ1 , σ2 , ς) 2   1 2 − σ 1 d1 − σ 2 ς + ης + d2 , 2

674

675

676

677

678 679 680 681

unc orre cted

664

where

(89)

G a (σ1 , σ2 , ς) = A + σ1 B1 + σ2 ςB2 , F(σ1 , σ2 , ς) = f − σ1 C1 − σ2 ςC2 . Example 4 We consider A to be a 2 × 2 diagonal matrix, i.e., a11 = −0.4, a12 = a21 = 0, and a22 = 0.6. Setting f = (0.3, −0.15)T , B1 = B2 = I, C1 = C2 = 0, d1 = 2, d2 = 1.2, and η = 1.7, the graph of the objective function P(x1 , x2 ) is a saddle surface (Fig. 4(left)), the constraint g1 (x1 , x2 ) = 21 (x12 + x22 ) ≤ 2 is a disk of radius 2, while

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(a)

(b)

Fig. 4 (a) Graph of P(x); (b) Contours of P(x), constraints g1 (x1 , x2 ) ≤ d1 (disk with radius R ≤ 2, dashed circle), and g2 (x1 , x2 ) ≤ d2 (annulus with radius 0.55 ≤ R ≤ 2.55) for Example 4

683

 2 g2 (x1 , x2 ) = 21 21 (x12 + x22 ) − 1.7 ≤ 1.2 represents an annulus (see Fig. 4 (right)). Solving the dual problem, we get

unc orre cted

682

σ¯ 1 = 0.5503198, σ¯ 2 = 0, and ς¯ = 0.3159349.

684

685

The primal solution is therefore

x¯ =

686

687



 1.9957445 , −0.1303985

which is located on the boundary g1 (x¯1 , x¯2 ) = 0, and

P(¯x) = −1.4097812 = P d (σ¯ 1 , σ¯ 2 , ς). ¯

688

690

Concrete applications of quadratic minimization with two nonconvex polynomial constraints are given in [28].

691

6 Concluding remarks and open problems

689

692 693 694 695 696 697 698 699

700 701 702 703 704 705

We have presented a detailed application of the canonical duality theory to the general differentiable nonconvex optimization problem. This problem arises in many real-world applications. Using the canonical dual transformation, a unified canonical dual problem was formulated with zero duality gap, which can be solved by well-developed nonlinear optimization methods. Both global and local optimizers can be identified by the triality theory. Insightful connections of this canonical duality with the classical Lagrangian duality have also been presented for two special applications in order to highlight the main constructs that enable the perfect duality results. Furthermore, these applications show how: (1) the n-dimensional nonconvex constrained problem (Pq ) in Sect. 3 can be reformulated as an m-dimensional concave maximization dual problem (Pqd ) over a convex space Sq+ with m < n (m = 1 in Example 1); (2) the nonconvex discrete integer programming problem (Pi ) in Sect. 4 can be converted to a concave maximization dual problem (Pid ) over a convex continuous space Si+ , which can be solved easily if Si+ is non-empty.

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709 710 711 712 713 714 715 716 717 718 719 720 721 722 723

of

Author Proof

708

Generally speaking, optimal solutions for constrained nonconvex minimization problems are usually KKT points located on the boundary of the feasible sets. Due to the lack of global optimality criteria, it is very difficult for direct methods and the classical Lagrangian relaxations to find global minimizers. Therefore, most of these problems are considered to be NP-hard (see [42,46]). However, by the canonical duality theory, these KKT points can be easily determined by the critical points of the canonical dual problems. The triality theory can be used to develop potentially powerful algorithms for solving these problems. The canonical duality theory concepts and methodologies presented in this article can be used and generalized to solve many other difficult problems in global optimization, nonconvex analysis and mechanics, network communication, and scientific computations. In general, so long as the geometrical operator  is chosen properly, the canonical dual transformation method can be used to formulate perfect dual problems and the triality theory can be used to establish useful theoretical results. If the convex dual feasible space Sc+ is not empty, the canonical dual max{P d (σ , ς) : (σ , ς ) ∈ Sc+ } can be solved easily by well-developed convex minimization techniques. As indicated in [21], the primal problem (P d ) could be NP-hard for the class of problems where the canonical dual function has no critical point in Sc+ . In this case, the primal problem (19) is equivalent to the following canonical dual minimal stationary problem:

pro

707

(Pgd ) : min sta{P d (σ , ς) : (σ , ς) ∈ Sc }.

724

725 726 727

unc orre cted

706

(90)

Since P d (σ , ς ) is usually nonconvex on Sc , to solve this minimal stationary problem could be a challenging task and many theoretical issues remain open. For further details and comprehensive applications of the canonical duality theory, we refer the reader to [4,10,21,24,26,45].

728

732

Acknowledgements The authors would like to thank the two anonymous referees and an associate editor for their detailed comments and insightful suggestions. In particular, Example 2 was suggested by one of reviewers and Remark 3 is based on the questions raised by this reviewer and the associate editor. This research was supported by the National Science Foundation under grants CCF-0514768 and CMMI-0552676.

733

References

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1. Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms, 3rd edn. Wiley, New York (2006) 2. Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976) 3. Fang, S.C., Gao, D.Y., Sheu, R.L., Wu, S.Y.: Canonical dual approach for solving 0-1 quadratic programming problems. J. Ind. Manag. Optim. 4(1), 125–142 (2008) 4. Fang, S.C., Gao, D.Y., Sheu, R.L., Xing, W.X.: Global optimization for a class of fractional programming problems. J. Glob. Optim., (2008) (in press) 5. Gao, D.Y.: Extended bounding theorems of limit analysis. Appl. Math. Mech. 4, 571–584 (1983) 6. Gao, D.Y.: Panpenalty finite element programming for limit analysis. Comput. Struct. 28, 749– 755 (1988) 7. Gao, D.Y.: Duality, triality and complementary extremun principles in nonconvex parametric variational problems with applications. IMA J. Appl. Math. 61, 199–235 (1998) 8. Gao, D.Y.: Pure complementary energy principle and triality theory in finite elasticity. Mech. Res. Comm. 26(1), 31–37 (1999) 9. Gao, D.Y.: General analytic solutions and complementary variational principles for large deformation nonsmooth mechanics. Meccanica 34, 169–198 (1999) 10. Gao, D.Y.: Duality Principles in Nonconvex Systems: Theory, Methods and Applications. pp. 454 Kluwer Academic Publishers, Dordrecht (2000) 11. Gao, D.Y.: Analytic solution and triality theory for nonconvex and nonsmooth vatiational problems with applications. Nonlinear Anal. 42(7), 1161–1193 (2000)

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759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813

of

Author Proof

758

pro

756 757

12. Gao, D.Y.: Canonical dual transformation method and generalized triality theory in nonsmooth global optimization. J. Glob. Optim. 17(1/4), 127–160 (2000) 13. Gao, D.Y.: Finite deformation beam models and triality theory in dynamical post-buckling analysis. Int. J. Non-Linear Mech. 5, 103–131 (2000) 14. Gao, D.Y.: Tri-duality in global optimization. In: Floudas, C.A., Pardalos, P.D. (eds.) Encyclopedia of Optimization, Vol. 1, pp. 485–491. Kluwer Academic Publishers, Dordrecht (2001) 15. Gao, D.Y.: Nonconvex Semi-linear Problems and Canonical Dual Solutions. Advances in Mechanics and Mathematics, Vol. II. pp. 261–312. Kluwer Academic Publishers, Dordrecht (2003) 16. Gao, D.Y.: Perfect duality theory and complete solutions to a class of global optimization problems. Optimization 52(4–5), 467–493 (2003) 17. Gao, D.Y.: Complete solutions to constrained quadratic optimization problems, Special issue on Duality. J. Glob. Optim. 29, 377–399 (2004) 18. Gao, D.Y.: Sufficient conditions and perfect duality in nonconvex minimization with inequality constraints. J. Ind. Manag. Optim. 1, 59–69 (2005) 19. Gao, D.Y.: Complete solutions and extremality criteria to polynomial optimization problems. J. Glob. Optim. 35, 131–143 (2006) 20. Gao, D.Y.: Solutions and optimality to box constrained nonconvex minimization problems. J. Ind. Maneg. Optim. 3(2), 293–304 (2007) 21. Gao, D.Y.: Advances in canonical duality theory with applications to global optimization. In: Ierapetriou, M., Bassett, M., Pistikopoulos, S. (eds.) Proceedings of the Fifth International Conference on Foundations of Computer-Aided Process Operations, pp.73–82. Omni Press, Cambridge, MA (2008) 22. Gao, D.Y., Ogden, R.W.: Closed-form solutions, extremality and nonsmoothness criteria in a large deformation elasticity problem. Zeitschrift für Angewandte Mathematik Und Physik 59(3), 498– 517 (2008) 23. Gao, D.Y., Ogden, R.W.: Multi-solutions to nonconvex variational problems with implications for phase transitions and numerical computation. Q. J. Mech. Appl. Math. 61(4), 497–522 (2008) 24. Gao, D.Y., Ruan, N.: Solutions and optimality criteria for nonconvex quadratic-exponential minimization problem. Math. Methods Oper. Res. 67(3), 479–496 (2008) 25. Gao, D.Y., Ruan, N.: On the solutions to quadratic minimization problems with box and integer constraints. J. Glob. Optim. (2008) (in press) 26. Gao, D.Y., Sherali, H.D.: Canonical Duality Theory: Connections Between Nonconvex Mechanics and Global Optimization, Advances in Applied Mathematics and Global Optimization. pp. 249– 316. Springer, New York (2008) 27. Gao, D.Y., Strang, G.: Geometric nonlinearity: potential energy, complementary energy, and the gap function. Q. Appl. Math. 47(3), 487–504 (1989) 28. Gao, D.Y., Yang, W.C.: Complete solutions to minimal distance problem between two nonconvex surfaces. Optimization 57(5) (2008) 29. Gao, D.Y., Yu, H.F.: Multi-scale modelling and canonical dual finite element method in phase transitions of solids. Int. J. Solids Struct. 45, 3660–3673 (2008) 30. Gao, D.Y., Ruan, N., Pardalos, P.M.: Canonical dual solutions to sum of fourth-order polynomials minimization problems with applications to sensor network localization, to appear. In: Pardalos, P.M., Ye, Y., Boginski, V.L., Commander, C.W. (eds.) Sensors: Theory, Algorithms, and Applications. Springer, Berlin (2008) 31. Hansen, P., Jaumard, B., Ruiz, M., Xiong, J.: Global minimization of indefinite quadratic functions subject to box constraints. Technical report, Technical Report G-91-54, Gread. École Polytechnique, Université McGill, Montreal (1991) 32. Hellinger, E.: Die allgemeine Ansätze der Mechanik der Kontinua. Encyklopädie der Mathematischen Wissenschaften IV(Part 4), 602–694 (1914) 33. Horst, R., Pardalos, P.M., Thoai, N.V.: Introduction to Global Optimization. Kluwer Academic Publishers, Dordrecht (2000) 34. Jeyakumar, V., Rubinov, A.M., Wu, Z.Y.: Sufficient global optimality conditions for non-convex quadratic minimization problems with box constraints. J. Glob. Optim. 36(3), 471–481 (2006) 35. Jeyakumar, V., Rubinov, A.M., Wu, Z.Y.: Non-convex quadratic minimization problems with quadratic constraints: global optimality conditions. Math. Program. 110(3), 521–541 (2007) 36. Johnson, D.E., Cohen, E.: A framework for efficient minimum distance computations. In: Proc. IEEE International Conference on Robotics and Automation, pp. 3678–3684. Leuven, Belgium (1998) 37. Koiter, W.T.: On the complementary energy theorem in nonlinear elasticity theory. In: G. Fichera (ed.) Trends Appl. Pure Math. Mech., Pitman, London (1976) 38. Lagrange, J.L.: Mécanique Analytique. Gauthier-Villars, Paris (1788) 39. Li, S.F., Gupta, A.: On dual configuration forces. J. Elast. 84, 13–31 (2006)

unc orre cted

754 755

123 Journal: JOGO MS: JOGO-07-692 CMS: 10898_2009_9399

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Author Proof

817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832

of

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40. Nash, S.G., Sofer, A.: Linear and Nonlinear Programming. McGraw-Hill, New York (1996) 41. Ogden, R.W.: A note on variational theorems in non-linear elastostatics. Math. Proc. Camb. Phil. Soc 77, 609–615 (1975) 42. Pardalos, P.M., Vavasis, S.A.: Quadratic programming with one negative eigenvalue is NP-hard. J. Glob. Optim. 21, 843–855 (1991) 43. Reissner, E.: On a variational theorem for finite elastic deformations. J. Math. Phys., 32(2–3), 129135 (1953). (See also Selected Works in Applied Mechanics and Mathematics, Jones and Bartlett Publishers, Boston, MA, 1996) 44. Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970) 45. Ruan, N., Gao, D.Y., Jiao, Y.: Canonical dual least square method for solving general nonlinear systems of quadratic equations. Comput. Optim. Appl. (published online: http://www.springerlink.com/ content/c6090221p4g41858/) (2008). DOI:10.1007/s10589-008-9222-5 46. Sahni, S.: Computationally related problems. SIAM J. Comp. 3, 262–279 (1974) 47. Sherali, H.D., Tuncbilek, C.: A global optimization for polynomial programming problem using a reformulation-linearization technique. J. Glob. Optim. 2, 101–112 (1992) 48. Sherali, H.D., Tuncbilek, C.: New reformulation-linearization technique based relaxation for univariate and multivariate polynomial programming problems. Oper. Res. Lett. 21(1), 1–10 (1997) 49. Wang, Z., Fang, S.C., Gao, D.Y., Xing, W.: Global extremal conditions for multi-integer quadratic programming. J. Ind. Manag. Optim. 4(2), 213–226 (2008)

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