On the Triality Theory for a Quartic Polynomial Optimization Problem
aa r X i v : . [ m a t h . O C ] O c t Manuscript submitted to Website: http://AIMsciences.orgAIMS’ JournalsVolume X , Number , XX pp. X–XX
ON THE TRIALITY THEORY FOR A QUARTIC POLYNOMIALOPTIMIZATION PROBLEM
David Yang Gao
School of Science, Information Technology and Engineering,University of Ballarat, Victoria 3353, Australia.
Changzhi Wu
School of Science, Information Technology and Engineering,University of Ballarat, Victoria 3353, Australia.School of Mathematics, Chongqing Normal University,Shapingba, Chongqing, 400047, China(Communicated by Kok Lay Teo)
Abstract.
This paper presents a detailed proof of the triality theorem for aclass of fourth-order polynomial optimization problems. The method is basedon linear algebra but it solves an open problem on the double-min duality leftin 2003. Results show that the triality theory holds strongly in a tri-dualityform if the primal problem and its canonical dual have the same dimension;otherwise, both the canonical min-max duality and the double-max dualitystill hold strongly, but the double-min duality holds weakly in a symmetricalform. Four numerical examples are presented to illustrate that this theory canbe used to identify not only the global minimum, but also the largest localminimum and local maximum.1.
Introduction and Motivation.
The concepts of triality and tri-duality were originally pro-posed in nonconvex mechanics [4, 5]. Mathematical theory of triality in its standard format iscomposed of three types of dualities: a canonical min-max duality and a pair of double-min anddouble-max dualities. The canonical min-max duality provides a sufficient condition for globalminimum, while the double-min and double max dualities can be used to identify respectively thelargest local minimum and local maximum. The tri-duality is a strong form of the triality princi-ple [7]. Together with a canonical dual transformation and a complementary-dual principle, theycomprise a versatile canonical duality theory, which can be used not only for solving a large classof challenging problems in nonconvex/nonsmooth analysis and continuous/discrete optimization[7, 8], but also for modeling complex systems and understanding multi-scale phenomena within aunified framework [4, 5, 7] (see also the review articles [10, 12, 17]).For example, in the recent work by Gao and Ogden [13] on nonconvex variational/boundaryvalue problems, it was discovered that both the global and local minimizers are usually nonsmoothfunctions and cannot be determined easily by traditional Newton-type numerical methods. How-ever, by the canonical dual transformation, the nonlinear differential equation is equivalent to analgebraic equation, which can be solved analytically to obtain all solutions. Both global mini-mizer and local extrema were identified by the triality theory, which revealed some interestingphenomena in phase transitions.The triality theory has attracted much attention recently in duality ways: Successful applica-tions in multi-disciplinary fields of mathematics, engineering and sciences show that this theory isnot only useful and versatile, but also beautiful in its mathematical format and rich in connotation2000
Mathematics Subject Classification.
Key words and phrases. canonical duality, triality, global optimization, polynomial optimiza-tion, counter-examples. of physics, which reveals a unified intrinsic duality pattern in complex systems; On the other hand,a large number of “counterexamples” have been presented in several papers since 2010. Unfortu-nately, most of these counterexamples are either fundamentally wrong (see [30, 31]), or repeatedlyaddress an open problem left by Gao in 2003 on the double-min duality [9, 10].The main goal of this paper is to solve this open problem left in 2003. The next section willpresent a brief review and the open problem in the triality theory. In Section 3, the trialitytheory is proved in its strong form as it was originally discovered. Section 4 shows that boththe canonical min-max and the double-max dualities hold strongly in general, but the double-minduality holds weakly in a symmetrical form. Applications are illustrated in Section 5, where alinear perturbation method is used for solving certain critical problems. The paper ended by anAppendix and a section of concluding remarks.2.
Canonical Duality Theory: A Brief Review and an Open Problem.
Let us begin withthe general global extremum problem( P ) : ext (cid:26) Π ( x ) = W ( x ) + 12 h x , A x i − h x , f i | x ∈ X a (cid:27) , (1)where X a ⊂ R n is an open set, x = { x i } ∈ R n is a decision vector, A = { A ij } ∈ R n × n is agiven symmetric matrix, f = { f i } ∈ R n is a given vector, and h∗ , ∗i denotes a bilinear form on R n × R n ; the function W : X a → R is assumed to be nonconvex and differentiable (it is allowed tobe nonsmooth and sub-differentiable for constrained problems). The notation ext {∗} stands forfinding global extremal of the function given in {∗} .In this paper, we are interested only in three types of global extrema: the global minimum anda pair of the largest local minimum and local maximum. Therefore, the nonconvex term W ( x ) in(1) is assumed to satisfy the objectivity condition , i.e., there exists a (geometrically) nonlinearmapping Λ : X a → V ⊂ R m and a canonical function V : V ⊂ R m → R such that W ( x ) = V (Λ ( x )) ∀ x ∈ X a . According to [7], a real valued function V : V → R is said to be a canonical function on its effectivedomain V a ⊂ V if its Legendre conjugate V ∗ : V ∗ → R V ∗ ( ς ) = sta {h ξ ; ς i − V( ξ ) | ξ ∈ V a } (2)is uniquely defined on its effective domain V ∗ a ⊂ V ∗ such that the canonical duality relations ς = ∇ V ( ξ ) ⇔ ξ = ∇ V ∗ ( ς ) ⇔ V ( ξ ) + V ∗ ( ς ) = h ξ ; ς i (3)hold on V a × V ∗ a , where h∗ ; ∗i represents a bilinear form which puts V and V ∗ in duality. Thenotation sta {∗} stands for solving the stationary point problem in {∗} . By this one-to-one canonicalduality, the nonconvex function W ( x ) = V (Λ( x )) can be replaced by h Λ( x ); ς i − V ∗ ( ς ) such thatthe nonconvex function Π( x ) in (1) can be written asΞ( x , ς ) = h Λ( x ); ς i − V ∗ ( ς ) + 12 h x , A x i − h x , f i , (4)which is the so-called total complementary (energy) function introduced by Gao and Strang in1989. By using this total complementary function, the canonical dual function Π d : V ∗ a → R canbe formulated as Π d ( ς ) = sta { Ξ( x , ς ) | ∀ x ∈ X a } = U Λ ( ς ) − V ∗ ( ς ) , (5)where U Λ : V ∗ a → R is called the Λ-conjugate of U ( x ) = h x , f i − h x , A x i , defined by [7] as U Λ ( ς ) = sta {h Λ( x ); ς i − U( x ) | x ∈ X a } . (6)Let S a ⊂ V ∗ a be the feasible domain of U Λ ( ς ); then the canonical dual problem is to solve thestationary point problem ( P d ) : ext { Π d ( ς ) | ς ∈ S a } . (7) The concept of objectivity in science means that qualitative and quantitative descriptionsof physical phenomena remain unchanged when the phenomena are observed under a variety ofconditions. That is, the objective function should be independent with the choice of the coordinatesystems. In continuum mechanics, the objectivity is also regarded as the principle of frame-indifference.
See Chapter 6 in [7] for mathematical definitions of the objectivity and geometricnonlinearity in differential geometry and finite deformation field theory. Detailed discussion ofobjectivity in global optimization will be given in another paper [18].
RIALITY THEORY FOR QUARTIC POLYNOMIAL OPTIMIZATION 3
Theorem 2.1 (Complementary-duality principle [7]) . Problem ( P d ) is a canonical dual to ( P ) in the sense that if (¯ x , ¯ ς ) is a critical point of Ξ( x , ς ) , then ¯ x is a critical point of ( P ) , ¯ ς is acritical point of ( P d ) , and Π(¯ x ) = Ξ(¯ x , ¯ ς ) = Π d (¯ ς ) . (8)Theorem 2.1 implies a perfect duality relation (i.e. no duality gap) between the primal problemand its canonical dual . The formulation of Π d ( ς ) depends on the geometrical operator Λ( x ). Inmany applications, the geometrical operator Λ is usually a quadratic mapping over a given field[7]. In finite dimensional space, this quadratic operator can be written as a vector-valued function(see [8], page 150) Λ( x ) = (cid:26) x T B k x (cid:27) mk =1 : X a ⊂ R n → V a ⊂ R m , (9)where B k = { B kij } ∈ R n × n is a symmetrical matrix for each k = 1 , , · · · , m , and V a ⊂ R m isdefined by V a = (cid:26) ξ ∈ R m | ξ k = 12 x T B k x ∀ x ∈ X a , k = 1 , . . . , m (cid:27) . In this case, the total complementary function has the formΞ ( x , ς ) = 12 h x , G ( ς ) x i − V ∗ ( ς ) − h x , f i , (10)where G : R m → R n × n is a matrix-valued function defined by G ( ς ) = A + m X k =1 ς k B k . (11)The critical condition ∇ x Ξ ( x , ς ) = 0 leads to the canonical equilibrium equation G ( ς ) x = f . (12)Clearly, for any given ς ∈ V ∗ a , if the vector f ∈ C ol ( G ( ς )), where C ol ( G ) stands for a spacespanned by the columns of G , the canonical equilibrium equation (12) can be solved analyticallyas x = [ G ( ς )] − f . Therefore, the canonical dual feasible space S a ⊂ V ∗ a can be defined as S a = { ς ∈ V ∗ a | f ∈ C ol ( G ( ς )) } , and on S a the canonical dual Π d ( ς ) is well-defined asΠ d ( ς ) = − h G ( ς ) − f , f i − V ∗ ( ς ) . (13) Theorem 2.2 (Analytic solution [7]) . If ¯ ς ∈ S a is a critical solution of ( P d ) , then ¯ x = G (¯ ς ) − f (14) is a critical solution of ( P ) and Π(¯ x ) = Π d (¯ ς ) .Conversely, if ¯ x is a critical solution of ( P ) , it must be in the form of (14) for a certain criticalsolution ¯ ς of ( P d ) . The canonical dual function Π d ( ς ) for a general quadratic operator Λ was first formulated innonconvex analysis, where Theorem 2.2 is called the pure complementary energy principle, [6].In finite deformation theory, this theorem solved an open problem left by Hellinger (1914) andReissner (1954) (see [25]). The analytical solution theorem has been successfully applied for solvinga class of nonconvex problems in mathematical physics, including Einstein’s special relativitytheory [7], nonconvex mechanics and phase transitions in solids [13]. In global optimization,the primal solutions to nonconvex minimization and integer programming problems are usuallylocated on the boundary of the feasible space. By Theorem 2.2, these solutions can be analyticallydetermined by critical points of the canonical dual function Π d ( ς ) (see [3, 11, 14, 16]). The complementary-dual in physics means perfect dual in optimization, i.e., the canonicaldual in Gao’s work, which means no duality gap. Otherwise, any duality gap will violet theenergy conservation law. Therefore, each complementary-dual variational statement in continuummechanics is usually refereed as a principle. In this paper G − should be understood as the generalized inverse if det G = 0 [8]. D. GAO AND C.Z. WU
In order to identify both global and local extrema of the primal and dual problems, we assume,without losing much generality, that the canonical function V : E a → R is convex and let S + a = { ς ∈ S a | G ( ς ) (cid:23) } , (15) S − a = { ς ∈ S a | G ( ς ) ≺ } , (16)where G ( ς ) (cid:23) G ( ς ) is a positive semi-definite matrix and G ( ς ) ≺ G ( ς ) is negative definite. Theorem 2.3 (Triality Theorem [8]) . Let (¯ x , ¯ ς ) be a critical point of Ξ ( x , ς ) .If G (¯ ς ) (cid:23) , then ¯ ς is a global maximizer of Problem ( P d ) , the vector ¯ x is a global minimizerof Problem ( P ) , and the following canonical min-max duality statement holds: min x ∈X a Π ( x ) = Ξ (¯ x , ¯ ς ) = max ς ∈S + a Π d ( ς ) . (17) If G (¯ ς ) ≺ , then there exists a neighborhood X o × S o ⊂ X a × S − a of (¯ x , ¯ ς ) for which we haveeither the double-min duality statement min x ∈X o Π ( x ) = Ξ (¯ x , ¯ ς ) = min ς ∈S o Π d ( ς ) , (18) or the double-max duality statement max x ∈X o Π ( x ) = Ξ (¯ x , ¯ ς ) = max ς ∈S o Π d ( ς ) . (19)The triality theory provides actual global extremum criteria for three types of solutions to thenonconvex problem ( P ): a global minimizer ¯ x (¯ ς ) if ¯ ς ∈ S + a and a pair of the largest-valued localextrema. In other words, ¯ x (¯ ς ) is the largest-valued local maximizer if ¯ ς ∈ S − a is a local maximizer;¯ x (¯ ς ) is the largest-valued local minimizer if ¯ ς ∈ S − a is a local minimizer. This pair of largest localextrema plays a critical role in nonconvex analysis of post-bifurcation and phase transitions. Remark 1 (Relation between Lagrangian Duality and Canonical Duality) . The main difference between the Lagrangian-type dualities (including the equivalent Fenchel-Moreau-Rockfellar dualities) and the canonical duality is the operator
Λ : X a → V a . In fact, if Λ is linear, the primal problem ( P ) is called geometrically linear in [7] and the total complementaryfunction Ξ( x , ς ) is simply the well-known Lagrangian and is denoted as L ( x , ς ) = h Λ x ; ς i − V ∗ ( ς ) − F ( x ) . (20) In convex (static) systems, F ( x ) = h x , f i is linear and L ( x , ς ) is a saddle function. Therefore,the well-known saddle min-max duality links a convex minimization problem ( P ) to a concavemaximization dual problem with linear constraint: max { Π ∗ ( ς ) = − V ∗ ( ς ) | Λ ∗ ς = f , ς ∈ V ∗ a } , (21) where Λ ∗ is the conjugate operator of Λ defined via h Λ x ; ς i = h x , Λ ∗ ς i . Using the Lagrangemultiplier x ∈ X a to relax the equality constraint, the Lagrangian L ( x , ς ) is obtained. By thefact that the (canonical) duality in convex static systems is unique, the saddle min-max dualityis refereed as the mono-duality in complex systems (see Chapter 1 in [7] ).Since the linear operator Λ can not change the convexity of W ( x ) = V (Λ x ) , the Lagrangianduality theory can be used mainly for convex problems. It is known that if W ( x ) is nonconvex, thenthe Lagrangian duality as well as the related Fenchel-Moreau-Rockafellar duality will produce theso-called duality gap. Comparing the canonical dual function Π d ( ς ) in (13) with the Lagrangiandual function Π ∗ ( ς ) in (21), we know that the duality gap is h G ( ς ) − f , f i .The canonical duality theory is based on the (geometrically) nonlinear mapping Λ : X a → V a and the canonical transformation W ( x ) = V (Λ( x )) . The total complementary function Ξ( x , ς ) is also known as the nonlinear or extended Lagrangian and is denoted by L ( x , ς ) due to thegeometric nonlinearity of Λ( x ) (see [7, 10] ). Relations between the canonical duality and theclassical Lagrangian duality are discussed in [16] . Remark 2 (Geometrical Nonlinearity and Complementary Gap Function) . The canonical min-max duality statement (17) was first proposed by Gao and Strang in nonconvex/nonsmooth analy-sis and mechanics in 1989 [19] , where Π( x ) = W ( x ) − F ( x ) is the so-called total potential energywith W representing the internal (or stored) energy and F the external energy. The geometricalnonlinearity is a standard terminology in finite deformation theory, which implies that the geo-metrical equation (or the configuration-strain relation) ξ = Λ( x ) is nonlinear. By definition inphysics, a function F ( x ) is called the external energy means that its (sub-)differential must be the RIALITY THEORY FOR QUARTIC POLYNOMIAL OPTIMIZATION 5 external force (or input) f . Therefore, in Gao and Strang’s work, the external energy should bea linear function(al) F ( x ) = h x , f i on its effective domain. In this case, the matrix G ( ς ) is aHessian of the so-called complementary gap function (i.e. the Gao-Strang gap function [19] ) G ap ( x , ς ) = h− Λ c ( x ); ς i , (22) where Λ c ( x ) = − x T B k x is called the complementary operator of a Gˆateauxdifferential Λ t ( x ) x = x T B k x of Λ( x ) [19] . Actually, in Gao and Strang’s original work, the canonical min-max dualitystatement holds in a general (weak) condition, i.e., G ap ( x , ¯ ς ) ≥ , ∀ x ∈ X a in field theory(corresponding to the strong condition G (¯ ς ) (cid:23) ). The related canonical duality theory has beengeneralized to nonconvex variational analysis of a large deformation (von Karman) plate (where A = ∆ [34] ), nonconvex (chaotic) dynamical systems (where A = ∆ − ∂ /∂t [10] ), and generalnonconvex constrained problems in global optimization. Since F ( x ) in these general applicationsis the quadratic function − h x , A x i + h x , f i , the Gao-Strang gap function (22) should be replacedby the generalized form G ap ( x , ς ) = h x , G ( ς ) x i (see the review article by Gao and Sherali [17] ).This gap function recovers the existing duality gap in traditional duality theories and provides asufficient global optimality condition for general nonconvex problems in both infinite and finitedimensional systems (see review articles [10, 17] ). By the fact that the geometrical mapping Λ in Gao and Strang’s work is a tensor-like operator, it has been realized recently that the popularsemi-definite programming method is actually a special application (where W ( x ) is a quadraticfunction) of the canonical min-max duality theory proposed in 1989 (see [14, 15] ).In a recent paper by Voisei and Zalinescu [31] , they unfortunately misunderstood some ba-sic terminologies in continuum physics, such as geometric nonlinearity, internal and externalenergies, and present “counterexamples” to the Gao-Strang theory based on certain “artificiallychosen” operators Λ( x ) and quadratic functions F ( x ) . Whereas in the stated contexts, the ge-ometrical operator Λ should be a canonical measure (Cauchy-Reimann type finite deformationoperator, see Chapter 6 in [7] ) and the external energy F ( x ) is typically a linear functional on itseffective domain; otherwise, its (sub)-differential will not be the external force. Interested readersare refereed by [18] for further discussion. Remark 3 (Double-Min Duality and Open Problem) . The double-min and double-max dualitystatements were discovered simultaneously in a post-buckling analysis of large deformed beammodel [4, 5] in 1996, where the finite strain measure Λ is a quadratic differential operator froma 2-D displacement field to a 2-D canonical strain field. Therefore, the triality theory was firstproposed in its strong form, i.e. the so-called tri-duality theory (see the next section). Later onwhen Gao was writing his duality book [7] , he realized that this pair of double-min and double-max dualities holds naturally in convex Hamilton systems. Accordingly, a bi-duality theorem wasproposed and proved for geometrically linear systems (where Λ is a linear operator; see Chapter2 in [7] ). Following this, the triality theory was naturally generalized to geometrically nonlinearsystems (nonlinear Λ ; see Chapter 3 in [7] ) with applications to global optimization problems [8] .However, it was discovered in 2003 that if n = m in the quadratic mapping (9), the double-minduality statement needs “certain additional constraints”. For the sake of mathematical rigor, thedouble-min duality was not included in the triality theory and these additional constraints wereleft as an open problem (see Remark 1 in [9] , also Theorem 3 and its Remark in a review articleby Gao [10] ). By the fact that the double-max duality is always true, the double-min duality wasstill included in the triality theory in the “either-or” form in many applications (see [12, 16] ).However, ignoring the open problem related to the “certain additional constraints” on the double-min duality statement has led to some misleading results. The goal of this paper is to solve this open problem by providing a simple proof of the trialitytheory based on linear algebra. To help understanding the intrinsic characteristics of the originalproblem and its canonical dual, we assume that the nonconvex objective function W ( x ) is a sumof fourth-order canonical polynomials W ( x ) = 12 m X k =1 β k (cid:18) x T B k x − d k (cid:19) , (23)where B k = n B kij o ∈ R n × n , k = 1 , · · · , m, are all symmetric matrices, β k > d k ∈ R ,k = 1 , · · · , m are given constants. This polynomial is actually a discretized form of the so-called double-well potential , first proposed by van der Waals in thermodynamics in 1895 (see [26]), whichis the mathematical model for natural phenomena of bifurcation and phase transitions in biology, D. GAO AND C.Z. WU chemistry, cosmology, continuum mechanics, material science, and quantum field theory, etc. (see[5, 20, 23, 24]). By using the quadratic geometrical operator Λ( x ) given by (9), the canonicalfunction V ( ξ ) = 12 ( ξ − d ) T β ( ξ − d ) (24)and its Legendre conjugate V ∗ ( ς ) = 12 ς T β − ς + ς T d (25)are quadratic functions, where β = Diag ( β k ) represents the diagonal matrix defined by the non-zero vector { β k } .In the following discussions, we assume that all the critical points of problem ( P ) are non-singular, i.e., if ∇ Π(¯ x ) = 0, then det ∇ Π(¯ x ) = 0 . (26)We will first prove that if n = m , the triality theorem holds in its strong form; otherwise, thetheorem holds in its weak form. Three numerical examples are used to illustrate the effectivenessand efficiency of the canonical duality theory.3. Strong Triality Theory for Quartic Polynomial Optimization: Tri-Duality Theo-rem.
We first consider the case m = n . For simplicity, we assume that β k = 1 in the followingdiscussion (otherwise, B k can be replaced by √ β k B k and d k is replaced by d k / √ β k ). In thiscase, the problem (1) is denoted as problem ( P ). Its canonical dual is( P d ) : ext (cid:26) Π d ( ς ) = − f T [ G ( ς )] − f − ς T ς − ς T d | ς ∈ S a ⊂ R n (cid:27) . (27) Theorem 3.1 (Tri-Duality Theorem) . Suppose that m = n , that the assumption (26) is satisfied, that ¯ ς is a critical point of Problem ( P d ) and that ¯ x = [ G (¯ ς )] − f .If ¯ ς ∈ S + a , then ¯ ς is a global maximizer of Problem ( P d ) in S + a if and only if ¯ x is a globalminimizer of Problem ( P ) , i.e., the following canonical min-max statement holds: Π(¯ x ) = min x ∈ R n Π ( x ) ⇐⇒ max ς ∈S + a Π d ( ς ) = Π d (¯ ς ) . (28) On the other hand, if ¯ ς ∈ S − a , then, there exists a neighborhood X o × S o ⊂ R n × S − a of (¯ x , ¯ ς ) ,such that either one of the following two statements holds.(A) The double-min duality statement Π(¯ x ) = min x ∈X o Π ( x ) ⇐⇒ min ς ∈S o Π d ( ς ) = Π d (¯ ς ) , (29) or (B) the double-max duality statement Π(¯ x ) = max x ∈X o Π ( x ) ⇐⇒ max ς ∈S o Π d ( ς ) = Π d (¯ ς ) . (30) Proof.
If ¯ ς is a critical point of the canonical dual problem ( P d ), the criticality condition ∇ Π d ( ς ) = 12 f T [ G ( ς )] − B [ G ( ς )] − f · · · f T [ G ( ς )] − B n [ G ( ς )] − f − ς − d = 0 ∈ R n (31)leads to ¯ ς = Λ(¯ x ). By the fact that ∇ Π(¯ x ) = G (¯ ς )¯ x − f = 0 ∈ R n , it follows that ¯ x = [ G (¯ ς )] − f is a critical point of Problem ( P ).To prove the validity of the canonical min-max statement (28), let ¯ ς be a critical point and¯ ς ∈ S + a . Since Π d ( ς ) is concave on S + a , the critical point ¯ ς ∈ S + a must be a global maximizer ofΠ d ( ς ) on S + a .On the other hand, by the convexity of V ( ξ ), we have V ( ξ ) − V (cid:0) ¯ ξ (cid:1) ≥ (cid:10) ξ − ¯ ξ ; ∇ V (cid:0) ¯ ξ (cid:1)(cid:11) = h ξ − ¯ ξ ; ¯ ς i . (32)Substituting ξ = Λ ( x ) and ¯ ξ = Λ (¯ x ) into (32), we obtain V (Λ ( x )) − V (Λ (¯ x )) ≥ h Λ ( x ) − Λ (¯ x ) ; ¯ ς i . This leads toΠ ( x ) − Π (¯ x ) ≥ h Λ ( x ) − Λ (¯ x ) ; ¯ ς i + 12 h x , A x i − h ¯ x , A ¯ x i − h x − ¯ x , f i , ∀ x ∈ R n . (33) RIALITY THEORY FOR QUARTIC POLYNOMIAL OPTIMIZATION 7
By the fact that ¯ ς = Λ (¯ x ) − d , (34)we have Π ( x ) − Π (¯ x ) ≥ h x , G (¯ ς ) x i − h ¯ x , G (¯ ς ) ¯ x i − h x − ¯ x , G (¯ ς ) ¯ x i . (35)For a fixed ¯ ς ∈ S + a , the convexity of the complementary gap function G ap ( x , ¯ ς ) = h x , G (¯ ς ) x i on X a leads to G ap ( x , ¯ ς ) − G ap (¯ x , ¯ ς ) ≥ h x − ¯ x , ∇ x G ap (¯ x , ¯ ς ) i = h x − ¯ x , G (¯ ς ) ¯ x i ∀ x ∈ R n . (36)Therefore, we haveΠ ( x ) − Π (¯ x ) ≥ h x − ¯ x , G (¯ ς ) ¯ x i − h x − ¯ x , G (¯ ς ) ¯ x i = 0 ∀ x ∈ R n . (37)This shows that ¯ x is a global minimizer of Problem ( P ). Since it is assumed that ¯ ς ∈ S + a , itfollows that (28) is satisfied.We move on to prove the double-min duality statement (29).Let ¯ ς be a critical point of Π d ( ς ) and ¯ ς ∈ S − a . It is easy to verify that ∇ Π ( x ) = n X k =1 (cid:18) x T B k x − d k (cid:19) B k x + A x − f , ∇ Π (¯ x ) = G (¯ ς ) + F (¯ x ) F (¯ x ) T , (38)where F ( x ) = (cid:2) B x , B x , · · · , B n x (cid:3) . In light of (31), ∇ Π d (¯ ς ) can be expressed in terms of ¯ x = [ G (¯ ς )] − f as follows: ∇ Π d (¯ ς ) = − F (¯ x ) T [ G (¯ ς )] − F (¯ x ) − I , where I is the identity matrix. If the critical point ¯ ς ∈ S − a is a local minimizer, we have ∇ Π d (¯ ς ) (cid:23) . This leads to − F (¯ x ) T [ G (¯ ς )] − F (¯ x ) (cid:23) I . (39)Therefore, − F (¯ x ) T [ G (¯ ς )] − F (¯ x ) is positive definite and F (¯ x ) is invertible. By multiplying (cid:16) F (¯ x ) T (cid:17) − and F (¯ x ) − to the left and right sides of (39), respectively, we obtain − [ G (¯ ς )] − (cid:23) (cid:16) F (¯ x ) T (cid:17) − ( F (¯ x )) − . According to Lemma 6.2 in Appendix, the following matrix inequality is obtained: ∇ Π (¯ x ) = G (¯ ς ) + F (¯ x ) F (¯ x ) T (cid:23) . By the assumption (26), ¯ x = [ G (¯ ς )] − f is also a local minimizer of Problem ( P ) . Therefore, ona neighborhood X o × S o ⊂ R n × S − a of (¯ x , ¯ ς ) , we havemin x ∈X o Π ( x ) = Ξ (¯ x , ¯ ς ) = min ς ∈S o Π d ( ς ) . Similarly, we can show that if ¯ x is a local minimizer of Problem ( P ) , the corresponding ¯ ς is alsoa local minimizer of Problem ( P d ) . The next task is to prove the double-max duality statement (30).Let ¯ ς ∈ S − a be a local maximizer of Problem ( P d ). Then, we have ∇ Π d (¯ ς ) (cid:22) . This gives us F (¯ x ) T [ G (¯ ς )] − F (¯ x ) + I (cid:23) . (40)Now we have two possible cases regarding the invertibility of F (¯ x ) . If F (¯ x ) is invertible, then byusing a similar argument as presented above, we can show that the relationsmax x ∈X o Π ( x ) = Ξ (¯ x , ¯ ς ) = max ς ∈S o Π d ( ς )hold on a neighborhood X o × S o ⊂ R n × S − a of (¯ x , ¯ ς ).If F (¯ x ) is not invertible, by Lemma 6.1 in the Appendix, there exists two orthogonal matrices E and K such that F (¯ x ) = EDK , (41)where E T E = I = K T K and D = Diag ( σ , · · · , σ r , , · · · ,
0) with σ ≥ σ ≥ · · · ≥ σ r > r = rank ( F (¯ x )) . Substituting (41) into (40), we obtain − K T D T E T [ G (¯ ς )] − EDK − I (cid:22) . (42) D. GAO AND C.Z. WU
Thus, − D T h E T G (¯ ς ) E i − D − I (cid:22) . (43)Applying Lemma 6.4 in Appendix to (43), it follows that E T G (¯ ς ) E + DD T = E T G (¯ ς ) E + DKK T D T (cid:22) . Finally, we have ∇ Π (¯ x ) = G (¯ ς ) + EDKK T D T E T = G (¯ ς ) + F (¯ x ) F (¯ x ) T (cid:22) . This means that ¯ x is also a local maximizer of Problem ( P ) under the assumption (26), i.e., thereexists a neighborhood X o × S o ⊂ R n × S − a of (¯ x , ¯ ς ) such thatmax x ∈X o Π ( x ) = Ξ (¯ x , ¯ ς ) = max ς ∈S o Π d ( ς ) . Finally, we can show, in a similar way, that if ¯ x = [ G (¯ ς )] − f is a local maximizer of Problem( P ) and ¯ ς ∈ S − a , the corresponding ¯ ς is also a local maximizer of Problem ( P d ) . Therefore, thetri-duality theorem is proved. (cid:4)
Remark 4.
The strong triality Theorem 3.1 can also be used to identify saddle points of the primalproblem, i.e. ¯ ς ∈ S − a is a saddle point of Π d ( ς ) if and only if ¯ x = G (¯ ς ) − f is a saddle pointof Π( x ) on X a . Since the saddle points do not produce computational difficulties in numericaloptimization, and do not exist physically in static systems, these points are excluded from thetriality theory. Remark 5.
By the proof of Theorem 3.1, we know that if there exists a critical point ¯ ς ∈ S − a suchthat ¯ ς is a local minimizer of Problem ( P d ), then F (¯ x ) must be invertible. On the other hand, ifthe symmetric matrices { B k } are linearly dependent, then F ( x ) is not invertible for any x ∈ R n .In this case, the corresponding canonical dual problem ( P d ) has no local minimizers in S − a , andfor any critical point ¯ ς ∈ S − a , the analytical solution ¯ x = [ G (¯ ς )] − f is not a local minimizer ofΠ( x ).4. Refined Triality Theory for General Quartic Polynomial Optimization.
Let us recallthe primal problem and its canonical dual problem in the general quartic polynomial case ( n = m ):( P ) : ext ( Π ( x ) = 12 m X k =1 (cid:18) x T B k x − d k (cid:19) + 12 x T A x − x T f | x ∈ R n ) , (44) (cid:16) P d (cid:17) : ext (cid:26) Π d ( ς ) = − f T [ G ( ς )] − f − ς T ς − ς T d | ς ∈ S a ⊂ R m (cid:27) . (45)Suppose that ¯ x and ¯ ς are the critical points of Problem ( P ) and Problem ( P d ) , respectively, where¯ x = [ G (¯ ς )] − f . It is easy to verify that ∇ Π (¯ x ) = G (¯ ς ) + F (¯ x ) F (¯ x ) T ∈ R n × n (46) ∇ Π d (¯ ς ) = − F (¯ x ) T [ G (¯ ς )] − F (¯ x ) − I ∈ R m × m . (47)In this case, F ( x ) = (cid:2) B x , B x , · · · , B m x (cid:3) ∈ R n × m . To continue, we show the following lemmas.
Lemma 4.1.
Suppose that m < n . Let the critical point ¯ ς ∈ S − a be a local minimizer of Π d ( ς ) ,and let ¯ x = [ G (¯ ς )] − f . Then, there exists a matrix P ∈ R n × m with rank( P ) = m such that P T ∇ Π (¯ x ) P (cid:23) . (48) Proof.
By the fact that the critical point ¯ ς ∈ S − a is a local minimizer of Π d ( ς ), we have ∇ Π d (¯ ς ) = 0and ∇ Π d (¯ ς ) (cid:23)
0. It follows that − F (¯ x ) T [ G (¯ ς )] − F (¯ x ) (cid:23) I ∈ R m × m . Thus, rank( F (¯ x )) = m. Since ¯ ς ∈ S − a and F (¯ x ) F (¯ x ) T (cid:23) , there exists a non-singular matrix T ∈ R n × n such that T T G (¯ ς ) T = Diag ( − λ , · · · , − λ n ) (49)and T T F (¯ x ) F (¯ x ) T T = Diag ( a , · · · , a m , , . . . , , (50)where λ i > , i = 1 , · · · , n, and a j > , j = 1 , · · · , m. RIALITY THEORY FOR QUARTIC POLYNOMIAL OPTIMIZATION 9
According to the singular value decomposition theory [22], there exist orthogonal matrices U and E such that T T F (¯ x ) = U √ a .. . √ a m · · · · · · · · · E . Therefore, U is an identity matrix. Let R = √ a . .. √ a m · · · · · · · · · . Then, ∇ Π d (¯ ς ) = − F (¯ x ) T [ G (¯ ς )] − F (¯ x ) − I = − (cid:16) F T T (cid:17) h T T G (¯ ς ) T i − T T F (¯ x ) − I = − E T RU T [Diag ( − λ , · · · , − λ n )] − URE − I ∈ R m × m . Since ∇ Π d (¯ ς ) (cid:23) U is an identity matrix, and E is an orthogonal matrix, we have − R [Diag ( λ , · · · , λ n )] − R − I m × m = Diag (cid:18) a λ − , · · · , a m λ m − (cid:19) (cid:23) . Thus, a i ≥ λ i , i = 1 , · · · , m. Note that T T ∇ Π (¯ x ) T = Diag ( a − λ , · · · , a m − λ m , − λ m +1 , · · · , − λ n ) . (51)Let J = [ I m × m , m × ( n − m ) ] T . Then, we have J T T T ∇ Π (¯ x ) TJ = Diag ( a − λ , · · · , a m − λ m ) (cid:23) . (52)Let P = TJ . Clearly, rank( P ) = m and P T ∇ Π (¯ x ) P = Diag ( a − λ , · · · , a m − λ m ) (cid:23) . Theproof is completed. (cid:4) In a similar way, we can prove the following lemma.
Lemma 4.2.
Suppose that m > n . Let ¯ x = [ G (¯ ς )] − f be a critical point, which is a localminimizer of Problem ( P ) , where ¯ ς ∈ S − a . Then, there exists a matrix Q ∈ R m × n with rank( Q ) =n such that Q T ∇ Π d (¯ ς ) Q (cid:23) . (53)Let p , · · · , p m be the m column vectors of P and let q , · · · , q n be the n column vectors of Q , respectively. Clearly, p , · · · , p m are m independent vectors and q , · · · , q n are n independentvectors. We introduce the following two subspaces X ♭ = { x ∈ R n | x = ¯ x + θ p + · · · + θ m p m , θ i ∈ R , i = 1 , · · · , m } , (54) S ♭ = { ς ∈ R m | ς = ¯ ς + ϑ q + · · · + ϑ n q n , ϑ i ∈ R , i = 1 , · · · , n } . (55) Theorem 4.3 (Refined Triality Theorem) . Suppose that the assumption (26) is satisfied. Let ¯ ς be a critical point of Π d ( ς ) and let ¯ x =[ G (¯ ς )] − f .If ¯ ς ∈ S + a , then the canonical min-max duality holds in the strong form: Π(¯ x ) = min x ∈ R n Π ( x ) ⇐⇒ max ς ∈S + a Π d ( ς ) = Π d (¯ ς ) . (56) If ¯ ς ∈ S − a , then there exists a neighborhood X o × S o ⊂ R n × S − a of (¯ x , ¯ ς ) such that thedouble-max duality holds in the strong form Π(¯ x ) = max x ∈X o Π ( x ) ⇐⇒ max ς ∈S o Π d ( ς ) = Π d (¯ ς ) . (57) However, the double-min duality statement holds conditionally in the following super-symmetricalforms.
1. If m < n and ¯ ς ∈ S − a is a local minimizer of Π d ( ς ) , then ¯ x = [ G (¯ ς )] − f is a saddle point of Π( x ) and the double-min duality holds weakly on X o ∩ X ♭ × S o , i.e. Π(¯ x ) = min x ∈X o ∩X ♭ Π ( x ) = min ς ∈S o Π d ( ς ) = Π d (¯ ς ); (58)
2. If m > n and ¯ x = [ G (¯ ς )] − f is a local minimizer of Π( x ) , then ¯ ς is a saddle point of Π d ( ς ) andthe double-min duality holds weakly on X o × S o ∩ S ♭ , i.e. Π(¯ x ) = min x ∈X o Π ( x ) = min ς ∈S o ∩S ♭ Π d ( ς ) = Π d (¯ ς ) . (59) Proof.
The proof of the statements (56) and (57) are similar to that given for the proof ofTheorem 3.1. Thus, it suffices to prove the double-min duality statements (58) and (59).Firstly, we suppose that m < n and ¯ ς is a local minimizer of Problem ( P d ). Define ϕ ( t , · · · , t m ) = Π(¯ x + t ¯ x + · · · + t m ¯ x m ) . (60)From (47), we obtain − F (¯ x ) T [ G (¯ ς )] − F (¯ x ) (cid:23) I ∈ R m × m . Thus, F (¯ x ) T [ G (¯ ς )] − F (¯ x ) is a non-singular matrix and rank ( F ( ¯ x )) = m < n . We claim that¯ x = [ G (¯ ς )] − f is not a local minimizer of Problem ( P ) . On a contrary, suppose that ¯ x is also alocal minimizer. Then, we have ∇ Π (¯ x ) = G (¯ ς ) + F (¯ x ) F (¯ x ) T (cid:23) . Thus, F (¯ x ) F (¯ x ) T (cid:23) − G (¯ ς ) . Since ¯ ς ∈ S − a and rank ( F ) = m , it is clear that n = rank ( G (¯ ς )) = rank (cid:16) F (¯ x ) F (¯ x ) T (cid:17) = m. This is a contradiction. Therefore, ¯ x = [ G (¯ ς )] − f is a saddle point of Problem ( P ).It is easy to verify that Π(¯ x ) = Π d (¯ ς ). Thus, to prove (58), it suffices to prove that ∈ R m isa local minimizer of the function ϕ ( t , · · · , t n ).It is easy to verify that ∇ ϕ (0 , · · · ,
0) = [( ∇ Π(¯ x )) T p , · · · , ( ∇ Π(¯ x )) T p m ] T = ∇ Π(¯ x )) T P = (61)and ∇ ϕ (0 , · · · ,
0) = P T ∇ Π (¯ x ) P . (62)In light of Lemma 4.1 and the assumption (26), it follows that ∈ R m is, indeed, a local minimizerof the function ϕ ( t , · · · , t m ).In a similar way, we can establish the case of m > n . The proof is completed. (cid:4) Remark 6.
Theorem 4.3 shows that both the canonical min-max and double-max duality state-ments hold strongly for general cases; the double-min duality holds strongly for n = m but weaklyfor n = m in a symmetrical form. The “certain additional conditions” are simply the intersection X o T X ♭ for m < n and S o T S ♭ for m > n . Therefore, the open problem left in 2003 [9, 10] issolved for the double-well potential function W ( x ).The triality theory has been challenged recently in a series of more than seven papers, see,for example, [31, 32]. In the first version of [32], Voisei and Zalinescu wrote: “we consider thatit is important to point out that the main results of this (triality) theory are false. This is doneby providing elementary counter-examples that lead to think that a correction of this theory isimpossible without falling into trivia”. It turns out that most of these counter-examples simply usethe double-well function W ( x ) with n = m . In fact, these counter-examples address the same typeof open problem for the double-min duality left unaddressed in [9, 10]. Indeed, by Theorem 4.3,we know that both the canonical min-max duality and the double-max duality hold strongly forthe general case n = m . However, based on the weak double-min duality, one can easily constructother V-Z type counterexamples , where the strong double-min duality holds conditionally when n = m . Also, interested readers should find that the references [9, 10] never been cited in any oneof their papers. RIALITY THEORY FOR QUARTIC POLYNOMIAL OPTIMIZATION 11 Numerical Experiments.
In this section, some simple numerical examples are presented toillustrate the canonical duality theory.
Example 1 ( m = n = 2 ). Let us first consider Problem ( P ) with n = m = 2.ext ( Π ( x ) = 12 "(cid:18) x T B x − d (cid:19) + (cid:18) x T B x − d (cid:19) + 12 x T A x − x T f | x ∈ R ) , (63)where A = (cid:20) a a (cid:21) , B = (cid:20) b
00 0 (cid:21) , B = (cid:20) b (cid:21) , f = [ f , f ] T . The canonical dual problem can be expressed asΠ d ( ς ) = − (cid:18) f a + ς b + f a + ς b (cid:19) − (cid:0) ς + ς (cid:1) − ( d ς + d ς ) . Thus, ∇ Π d ( ς ) = b f a + ς b ) − ς − b f a + ς b ) − ς − and ∇ Π d ( ς ) = (cid:20) − b f ( a + ς b ) − − − b f ( a + ς b ) − − (cid:21) . Now, we take b = b = f = f = 1, d = d = 1 and a = − , a = −
3. It is easy tocheck that Π d ( ς ) has only one critical point ¯ ς = (cid:16) √ , . (cid:17) in S + a and four criticalpoints ¯ ς = (3 / , . , ¯ ς = (cid:16) − √ , − . (cid:17) , ¯ ς = (cid:16) − √ , . (cid:17) , ¯ ς =(3 / , − . S − a , respectively. Furthermore, ¯ ς is a local minimizer and ¯ ς is a localmaximizer; the solutions ¯ ς and ¯ ς are saddle points of Π d ( ς ) in S − a . Thus, by Theorem 3.1, weknow that ¯ x = (2 . , . x = ( − , − . x = ( − . , − . x ) = − . < Π (¯ x ) = − . < Π (¯ x ) = 0 . . The graph of Π ( x ) and its contour are depicted in Figure 1. -4 -2 0 2 4 -4 -2 0 2 4-1001020-4 -2 0 2 4 -4 -2 0 2 4-4-2024 Figure 1.
Graph of Π( x ) (left) and contours of Π( x ) (right) forExample 1 Example 2 ( n = 2 , m = 1 ). We now consider Problem ( P ) with n = 2 , m = 1, A =Diag ( − . , − . B = Diag (1 , f = (0 . , . T , d = 4. Then, its dual problem isΠ d ( ς ) = − (cid:18) . − . ς + 0 . − . ς (cid:19) − ς − ς. We can verify that Π d ( ς ) has one critical point ¯ ς = − . S + a and two critical points ¯ ς = − . ς = − . S − a . Furthermore, ¯ ς is a local minimizer and ¯ ς is a localmaximizer of Π d ( ς ). According to Theorem 4.3, ¯ x = ( A + ¯ ς B ) − f = (1 . , . T is the unique global minimizer of Π( x ), ¯ x = ( A + ¯ ς B ) − f = ( − . , − . T is a saddle point and ¯ x = ( A + ¯ ς B ) − f = ( − . , − . T is a local maximizerof Π( x ). Let p = (1 , T and ϕ ( θ ) = Π(¯ x + θ p ). Then, it is easy to verify that there existsneighborhoods X ⊂ R and S ⊂ R such that 0 ∈ X , ¯ ς ∈ S andmin θ ∈X ϕ ( θ ) = min ς ∈S Π d ( ς ) . This example shows that even if n > m , the canonical min-max duality and the double-maxduality still hold strongly. However, the double-min duality statement should be refined into an m − dimensional subspace in this case. Example 3 ( n = 1 , m = 2 ). We now consider Problem ( P ) with n = 1 , m = 2, A = − . B = 0 . B = 0 . d = 3, d = 2 . f = 1 .
4. Then, its dual problem isΠ d ( ς ) = − (cid:18) f A + ς B + ς B (cid:19) −
12 ( ς + ς ) − ( d ς + d ς ) . We can verify that Π d ( ς ) has one critical point ¯ ς = ( − . , . T in S + a and twocritical points ¯ ς = ( − . , − . T and ¯ ς = ( − . , . T in S − a .Furthermore, ¯ ς is a local maximizer and ¯ ς is a saddle point of Π d ( ς ). According to Theorem4.3, ¯ x = G (¯ ς ) − f = 4 . x ), ¯ x = G (¯ ς ) − f = − . x ). Let q = (1 , T and ψ ( ϑ ) = Π d (¯ ς + ϑ q ). Then, it iseasy to verify that there exists neighborhoods X ⊂ R and S ⊂ R such that ¯ x ∈ X , 0 ∈ S andmin x ∈X Π( x ) = min ϑ ∈S ψ ( ϑ ) . This example shows that if n < m , the canonical min-max duality and the double-max dualitystill hold strongly. However, the double-min duality statement should be refined into an n − dimensional subspace in this case. Example 4 Linear Perturbation.
Let us consider the following optimization problem withoutinput ( f = 0)( P ) : ext ( Π ( x ) = 12 "(cid:18)
12 (x + x ) − (cid:19) + (cid:18)
12 (x − x ) − (cid:19) | x ∈ R ) . Problem ( P ) has four global minimizers ¯ x = (1 , , ¯ x = (0 , − , ¯ x = (0 , , ¯ x = ( − ,
0) andthe optimal cost value is 0 . Its canonical dual problem isΠ d ( ς ) = − (cid:0) ς + ς (cid:1) − ( ς + ς ) . Π d ( ς ) has only one critical point ¯ ς = (cid:0) − , − (cid:1) ∈ S − a . Furthermore, we can check that ¯ x =[ G (¯ ς )] − f = (0 ,
0) is a local maximizer of Problem ( P ) . Thus, we cannot use the canonical dualtransformation method to obtain the global minimizer of Problem ( P ) since this problem is in aperfect symmetrical form without input, which allows more than one global minimizer. Now weperturb Problem ( P ) as follows. (cid:16) P b (cid:17) : ext x ∈ R Π ( x ) = 12 "(cid:18)
12 (x + x ) − (cid:19) + (cid:18)
12 (x − x ) − (cid:19) − (x f + x f ) . Its canonical dual function is expressed asΠ d ( ς ) = − ς ς (cid:2) ( ς + ς ) (cid:0) f + f (cid:1) + 2 ( ς − ς ) f f (cid:3) − (cid:0) ς + ς (cid:1) −
12 ( ς + ς ) . Taking f = 0 . f = 0 .
005 and solving ∇ Π d (¯ ς ) = 0 , the results obtained are listed in Table1. We can see that ¯ ς = (0 . , . ∈ S + a and Π d ( ς ) = − . x = [ G (¯ ς )] − f = (0 . , . (cid:0) P b (cid:1) . Clearly, this¯ x is very close to ¯ x . If we take f = 0 . f = − . , the global minimizer of Problem (cid:0) P b (cid:1) is ¯ x = [ G (¯ ς )] − f = (0 . , − . x = (0 , − S + a , a linear perturbation could be usedto solve the primal problem. RIALITY THEORY FOR QUARTIC POLYNOMIAL OPTIMIZATION 13
Table 1.
Numerical results for Example 3 ς = ( ς , ς ) x = ( x , x ) G ( ς ) The primal problem ( − . , − . − . , − . G ≺ local max ( − . , − . − . , − . G ≺ saddle point (0 . , − . − . , − . G ≺ saddle point ( − . , − . . , − . G ≺ saddle point ( − . , − . . , − . G ≺ local min (0 . , − . . , . indefinite saddle point ( − . , . − . , . indefinite local max ( − . , . − . , . indefinite saddle point (0 . , . . , . G ≻ global min Appendix.
In this Appendix, we present several lemmas which are needed for the proofs ofTheorem 3.1 and Theorem 4.3.
Lemma 6.1 (Singular value decomposition [22]) . For any given G ∈ R n × n with rank( G ) = r , there exist U ∈ R n × n , D ∈ R n × n and R ∈ R n × n such that G = UDR , where U , R are orthogonal matrices, i.e., U T U = I = R T R , and D = Diag ( σ , · · · , σ r , , · · · , ,σ ≥ σ ≥ · · · ≥ σ r > . Lemma 6.2.
Suppose that G and U are positive definite. Then, G (cid:23) U if and only if U − (cid:23) G − . Proof.
The proof is trivial and is omitted here.
Lemma 6.3 (Proposition 2.1 in [21]) . For any given symmetric matrix M expressed in the form M = (cid:20) M M M M (cid:21) such that M ≻ . Then, M (cid:23) if and only if M − M M − M (cid:23) . The following lemma plays a key role in the proof of Theorem 3.1 and Theorem 4.3.
Lemma 6.4.
Suppose that P ∈ R n × n , U ∈ R m × m , and D ∈ R n × m . Furthermore, D = (cid:20) D r × ( m − r ) ( n − r ) × r ( n − r ) × ( m − r ) (cid:21) ∈ R n × n , where D ∈ R r × r is nonsingular, r = rank( D ) , and P = (cid:20) P P P P (cid:21) ≺ , U = (cid:20) U r × ( m − r ) ( m − r ) × r U (cid:21) ≻ , P ij and U ii , i, j = 1 , , are of appropriate dimension matrices. Then, P + DUD T (cid:22) ⇐⇒ − D T P − D − U − (cid:22) . (64) Proof.
Suppose that P + DUD T (cid:22) . Then, − P − DUD T = (cid:20) − P − D U D T − P − P − P (cid:21) (cid:23) . Since P = (cid:20) P P P P (cid:21) ≺ , it follows that − P ≻ . By Lemma 6.3, we have the followinginequality − P − D U D T + P P − P (cid:23) which leads to − P + P P − P (cid:23) D U D T . Since P ≺ U ≻ , it follows from Lemma 6.2 that (cid:16) − P + P P − P (cid:17) − (cid:22) (cid:16) D T (cid:17) − U − D − . Thus, D T (cid:16) − P + P P − P (cid:17) − D (cid:22) U − . (66)Note that P − = (cid:16) P − P P − P (cid:17) − P − P (cid:16) P P − P − P (cid:17) − (cid:16) P P − P − P (cid:17) − P P − (cid:16) P − P P − P (cid:17) . By virtue of Lemma 6.3, we obtain − (cid:20) D
00 0 (cid:21) T P − (cid:20) D
00 0 (cid:21) (cid:22) (cid:20) U −
00 U − (cid:21) = U − , i.e., the right hand side of (64) holds.In a similar way, we can show that if − D T P − D − U − (cid:22) , then P + DUD T (cid:22) . The proofis thus completed. (cid:4) Conclusion Remarks.
In this paper, we presented a rigorous proof of the double-min dualityin the triality theory for a quartic polynomial optimization problem based on elementary linearalgebra. Our results show that under some proper assumptions, the triality theory for a classof quartic polynomial optimization problems holds strongly in the tri-duality form if the primalproblem and its canonical dual have the same dimension. Otherwise, both the canonical min-maxand the double-max still hold strongly, but the double-min duality holds weakly in a symmetricform.
Acknowledgments.
The main results of this paper were announced in the 2nd World Congressof Global Optimization, July 3-7, 2011, Chania, Greece. The authors are sincerely indebtedto Professor Hanif Sherali at Virginia Tech for his valuable comments and suggestions. DavidGao’s research is supported by US Air Force Office of Scientific Research under the grant AFOSRFA9550-10-1-0487. Changzhi Wu was supported by National Natural Science Foundation of Chinaunder the grant
REFERENCES [1] K.T. Andrews, M.F. M’Bengue and M. Shillor,
Vibrations of a nonlinear dynamic beambetween two stops , Discrete and Continuous Dynamical Systems - Series B, 12(1), (2009), 23- 38.[2] I. Ekeland, and R. Temam, “Convex Analysis and Variational Problems”, North-Holland,1976.[3] S.C. Fang, D.Y. Gao, R.L. Sheu and S.Y. Wu,
Canonical dual approach for solving 0-1quadratic programming problems , J. Ind. and Manag. Optim. 4, (2008), 125-142.[4] D.Y. Gao,
Post-buckling analysis and anomalous dual variational problems in nonlinear beamtheory in “Applied Mechanics in Americans Proc. of the Fifth Pan American Congress ofApplied Mechanics”, L.A. Godoy, L.E. Suarez (Eds.), Vol. 4. The University of Iowa, Iowacity, August (1996).[5] D.Y. Gao,
Dual extremum principles in finite deformation theory with applications to post-buckling analysis of extended nonlinear beam theory , Applied Mechanics Reviews, 50 (11),(1997), S64-S71.[6] D.Y. Gao,
General analytic solutions and complementary variational principles for largedeformation nonsmooth mechanics , Meccanica 34, (1999), 169–198.[7] D.Y. Gao, “Duality Principles in Nonconvex Systems: Theory, Methods and Applications”,Kluwer Academic, Dordrecht, 2000.
RIALITY THEORY FOR QUARTIC POLYNOMIAL OPTIMIZATION 15 [8] D.Y. Gao,
Canonical dual transformation method and generalized triality theory in nonsmoothglobal optimization , J. Glob. Optim., 17(1), (2000), 127–160.[9] D.Y. Gao,
Perfect duality theory and complete solutions to a class of global optimizationproblems , Optim., 52(4–5), (2003), 467–493.[10] D.Y. Gao,
Nonconvex semi-linear problems and canonical dual solutions , In: Gao, D.Y.,Ogden, R.W. (eds.)
Advances in Mechanics and Mathematics, vol. II, pp. 261312. KluwerAcademic, Dordrecht (2003).[11] D.Y. Gao,
Solutions and optimality to box constrained nonconvex minimization problems , J.Ind. Manag. Optim. 3(2), (2007), 293–304.[12] D.Y. Gao,
Canonical duality theory: theory, method, and applications in global optimization ,Comput. Chem., 33, (2009), 1964-1972.[13] D.Y. Gao and R.W. Ogden,
Multiple solutions to non-convex variational problems with im-plications for phase transitions and numerical computation , Quart. J. Mech. Appl. Math. 61(4), (2008), 497-522.[14] D.Y. Gao and N. Ruan, Solutions to quadratic minimization problems with box and integerconstraints, J. Glob. Optim. 47(3), (2010), 463-484.[15] D.Y. Gao, N. Ruan,and P.M. Pardalos,
Canonical dual solutions to sum of fourth-order poly-nomials minimization problems with applications to sensor network localization , in
Sensors:Theory, Algorithms and Applications , P.M. Pardalos, Y.Y. Ye, V. Boginski, and C. Comman-der (eds). Springer, 2010.[16] D.Y. Gao, N. Ruan, and H. Sherali,
Solutions and optimality criteria for nonconvex con-strained global optimization problems with connections between canonical and Lagrangianduality , J. Glob. Optim. 45(3), (2009), 473-497.[17] D.Y. Gao and H.D. Sherali,
Canonical duality: Connection between nonconvex mechanicsand global optimization , in
Advances in Appl. Mathematics and Global Optimization , 249-316, Springer (2009).[18] D.Y. Gao, H.D. Sherali, and G. Strang,
Canonical duality: Objectivity, Triality, and Gapfunctions , in preparation.[19] D.Y. Gao and G. Strang,
Geometric nonlinearity: Potential energy, complementary energy,and the gap function , Quart. Appl. Math., 47(3), (1989), 487–504.[20] D.Y. Gao, and H.F. Yu,
Multi-scale modelling and canonical dual finite element method inphase transitions of solids , International Journal of Solids and Structures, 45, (2008), 3660–3673.[21] J. Gallier,
The Schur complement and symmetric positive semidefinite (and definite) matri-ces
Constructive quantum field theory , Mathematical Physics 2000, 111-127, Edited byA. Fokas, A. Grigoryan, T. Kibble, and B. Zegarlinski, Imperial College Press, London 2000.[24] T.W.B. Kibble,
Phase transitions and topological defects in the early universe , Aust. J. Phys.,50, (1997), 697-722.[25] S.F. Li and A. Gupta,
On dual configuration forces , J. of Elasticity, 84, (2006),13-31.[26] J.S. Rowlinson,
Translation of J.D. van der Waals’ “The thermodynamic theory of capillarityunder the hypothesis of a continuous variation of density” , J. Statist. Phys., 20, (1979), 197-244.[27] N. Ruan, D.Y. Gao, and Y. Jiao
Canonical dual least square method for solving generalnonlinear systems of equations , Comput. Optim. Appl. 47 (2), (2010), 335-347.[28] D.M.M. Silva, and D.Y. Gao,
Complete solutions and triality theory to a nonconvex opti-mization problem with double-well potential in R n , to appear in J. Math. Analy. Appl.[29] G. Strang, G.: Introduction to Applied Mathematics, Wellesley-Cambridge Press, 1986, 758pp.[30] R. Strugariu, M.D. Voisei, and C. Zalinescu, Counter-examples in bi-duality, triality andtri-duality
Some remarks concerning Gao-Strang’s complementary gapfunction , Applicable Analysis, (2010). DOI: 10.1080/00036811.2010.483427.[32] M.D. Voisei, and C. Zalinescu,
Counterexamples to some triality tri-duality results , J. Glob.Optim., DOI 10.1007/s10898-010-9592-y.[33] Z.B. Wang, S.C. Fang, D.Y. Gao,and W.X. Xing,
Canonical dual approach to solving themaximum cut problem , to appear in
J. Glob. Optim. [34] S.T. Yau, and D.Y. Gao,
Obstacle problem for von Karman equations , Adv. Appl. Math., 13,(1992), 123-141.
E-mail address : [email protected] E-mail address ::