On the Triality Theory for a Quartic Polynomial Optimization Problem

This paper presents a detailed proof of the triality theorem for a class of fourth-order polynomial optimization problems. The method is based on linear algebra but it solves an open problem on the double-min duality left in 2003. Results show that the triality theory holds strongly in a tri-duality form if the primal problem and its canonical dual have the same dimension; otherwise, both the canonical min-max duality and the double-max duality still hold strongly, but the double-min duality holds weakly in a symmetrical form. Four numerical examples are presented to illustrate that this theory can be used to identify not only the global minimum, but also the largest local minimum and local maximum.

1. Introduction and Motivation. The concepts of triality and tri-duality were originally proposed in nonconvex mechanics [4,5]. Mathematical theory of triality in its standard format is composed of three types of dualities: a canonical min-max duality and a pair of double-min and double-max dualities. The canonical min-max duality provides a sufficient condition for global minimum, while the double-min and double max dualities can be used to identify respectively the largest local minimum and local maximum. The tri-duality is a strong form of the triality principle [7]. Together with a canonical dual transformation and a complementary-dual principle, they comprise a versatile canonical duality theory, which can be used not only for solving a large class of challenging problems in nonconvex/nonsmooth analysis and continuous/discrete optimization [7,8], but also for modeling complex systems and understanding multi-scale phenomena within a unified 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/boundary value problems, it was discovered that both the global and local minimizers are usually nonsmooth functions and cannot be determined easily by traditional Newton-type numerical methods. However, by the canonical dual transformation, the nonlinear differential equation is equivalent to an algebraic equation, which can be solved analytically to obtain all solutions. Both global minimizer and local extrema were identified by the triality theory, which revealed some interesting phenomena in phase transitions.
The triality theory has attracted much attention recently in duality ways: Successful applications in multi-disciplinary fields of mathematics, engineering and sciences show that this theory is not only useful and versatile, but also beautiful in its mathematical format and rich in connotation  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. Unfortunately, most of these counterexamples are either fundamentally wrong (see [30,31]), or repeatedly address 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 will present a brief review and the open problem in the triality theory. In Section 3, the triality theory is proved in its strong form as it was originally discovered. Section 4 shows that both the canonical min-max and the double-max dualities hold strongly in general, but the double-min duality holds weakly in a symmetrical form. Applications are illustrated in Section 5, where a linear perturbation method is used for solving certain critical problems. The paper ended by an Appendix and a section of concluding remarks.
2. Canonical Duality Theory: A Brief Review and an Open Problem. Let us begin with the general global extremum problem where Xa ⊂ R n is an open set, x = {x i } ∈ R n is a decision vector, A = {A ij } ∈ R n×n is a given symmetric matrix, f = {f i } ∈ R n is a given vector, and * , * denotes a bilinear form on R n × R n ; the function W : Xa → R is assumed to be nonconvex and differentiable (it is allowed to be nonsmooth and sub-differentiable for constrained problems). The notation ext{ * } stands for finding global extremal of the function given in { * }.
In this paper, we are interested only in three types of global extrema: the global minimum and a pair of the largest local minimum and local maximum. Therefore, the nonconvex term W (x) in (1) is assumed to satisfy the objectivity condition 1 , i.e., there exists a (geometrically) nonlinear mapping Λ : Xa → V ⊂ R m and a canonical function According to [7], a real valued function V : V → R is said to be a canonical function on its effective domain Va ⊂ V if its Legendre conjugate V * : is uniquely defined on its effective domain V * a ⊂ V * such that the canonical duality relations hold on Va × V * a , where * ; * represents a bilinear form which puts V and V * in duality. The notation sta{ * } stands for solving the stationary point problem in { * }. By this one-to-one canonical duality, the nonconvex function W (x) = V (Λ(x)) can be replaced by Λ(x); ς − V * (ς) such that the nonconvex function Π(x) in (1) can be written as which is the so-called total complementary (energy) function introduced by Gao and Strang in 1989. By using this total complementary function, the canonical dual function Π d : V * a → R can be formulated as where Let Sa ⊂ V * a be the feasible domain of U Λ (ς); then the canonical dual problem is to solve the stationary point problem 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), thenx is a critical point of (P),ς is a critical point of (P d ), and Theorem 2.1 implies a perfect duality relation (i.e. no duality gap) between the primal problem and its canonical dual 2 . The formulation of Π d (ς) depends on the geometrical operator Λ(x). In many 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) where B k = {B k ij } ∈ R n×n is a symmetrical matrix for each k = 1, 2, · · · , m, and Va ⊂ R m is defined by In this case, the total complementary function has the form where G : R m → R n×n is a matrix-valued function defined by The critical condition ∇xΞ (x, ς) = 0 leads to the canonical equilibrium equation Clearly, for any given ς ∈ V * a , if the vector f ∈ C ol (G (ς)), where C ol (G) stands for a space spanned by the columns of G, the canonical equilibrium equation (12) can be solved analytically as 3 x = [G(ς)] −1 f . Therefore, the canonical dual feasible space Sa ⊂ V * a can be defined as Sa = {ς ∈ V * a | f ∈ C ol (G (ς))} , and on Sa the canonical dual Π d (ς) is well-defined as Theorem 2.2 (Analytic solution [7]). Ifς ∈ Sa is a critical solution of (P d ), then is a critical solution of (P) and Π(x) = Π d (ς). Conversely, ifx is a critical solution of (P), it must be in the form of (14) for a certain critical solutionς of (P d ).
The canonical dual function Π d (ς) for a general quadratic operator Λ was first formulated in nonconvex 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) and Reissner (1954) (see [25]). The analytical solution theorem has been successfully applied for solving a class of nonconvex problems in mathematical physics, including Einstein's special relativity theory [7], nonconvex mechanics and phase transitions in solids [13]. In global optimization, the primal solutions to nonconvex minimization and integer programming problems are usually located on the boundary of the feasible space. By Theorem 2.2, these solutions can be analytically determined by critical points of the canonical dual function Π d (ς) (see [3,11,14,16]). 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 : Ea → R is convex and let where G (ς) 0 means that G (ς) is a positive semi-definite matrix and G (ς) ≺ 0 means that 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) 0, thenς is a global maximizer of Problem (P d ), the vectorx is a global minimizer of Problem (P), and the following canonical min-max duality statement holds: If G(ς) ≺ 0, then there exists a neighborhood Xo × So ⊂ Xa × S − a of (x,ς) for which we have either the double-min duality statement or the double-max duality statement The triality theory provides actual global extremum criteria for three types of solutions to the nonconvex problem (P): a global minimizerx(ς) ifς ∈ S + a and a pair of the largest-valued local extrema. In other words,x(ς) is the largest-valued local maximizer ifς ∈ S − a is a local maximizer; x(ς) is the largest-valued local minimizer ifς ∈ S − a is a local minimizer. This pair of largest local extrema 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 : Xa → Va. In fact, if Λ is linear, the primal problem (P) is called geometrically linear in [7] and the total complementary function Ξ(x, ς) is simply the well-known Lagrangian and is denoted as In convex (static) systems, F (x) = x, f 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 concave maximization dual problem with linear constraint: where Λ * is the conjugate operator of Λ defined via Λx; ς = x, Λ * ς . Using the Lagrange multiplier x ∈ Xa to relax the equality constraint, the Lagrangian L(x, ς) is obtained. By the fact that the (canonical) duality in convex static systems is unique, the saddle min-max duality is 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 (Λx), the Lagrangian duality theory can be used mainly for convex problems. It is known that if W (x) is nonconvex, then the Lagrangian duality as well as the related Fenchel-Moreau-Rockafellar duality will produce the so-called duality gap. Comparing the canonical dual function Π d (ς) in (13) with the Lagrangian dual function Π * (ς) in (21), we know that the duality gap is 1 2 G(ς) −1 f , f . The canonical duality theory is based on the (geometrically) nonlinear mapping Λ : Xa → Va 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 the geometric nonlinearity of Λ(x) (see [7,10]). Relations between the canonical duality and the classical Lagrangian duality are discussed in [16].

Remark 2 (Geometrical Nonlinearity and Complementary Gap Function).
The canonical minmax duality statement (17) was first proposed by Gao and Strang in nonconvex/nonsmooth analysis and mechanics in 1989 [19], where Π(x) = W (x) − F (x) is the so-called total potential energy with W representing the internal (or stored) energy and F the external energy. The geometrical nonlinearity is a standard terminology in finite deformation theory, which implies that the geometrical equation (or the configuration-strain relation) ξ = Λ(x) is nonlinear. By definition in physics, a function F (x) is called the external energy means that its (sub-)differential must be the external force (or input) f . Therefore, in Gao and Strang's work, the external energy should be a linear function(al) F (x) = x, f on its effective domain. In this case, the matrix G(ς) is a Hessian of the so-called complementary gap function (i.e. the Gao-Strang gap function [19]) where [19]. Actually, in Gao and Strang's original work, the canonical min-max duality statement holds in a general (weak) condition, i.e., Gap(x,ς) ≥ 0, ∀x ∈ Xa in field theory (corresponding to the strong condition G(ς) 0). The related canonical duality theory has been generalized to nonconvex variational analysis of a large deformation (von Karman) plate (where , and general nonconvex constrained problems in global optimization. Since F (x) in these general applications is the quadratic function − 1 2 x, Ax + x, f , the Gao-Strang gap function (22) should be replaced by the generalized form Gap(x, ς) = 1 2 x, G(ς)x (see the review article by Gao and Sherali [17]). This gap function recovers the existing duality gap in traditional duality theories and provides a sufficient global optimality condition for general nonconvex problems in both infinite and finite dimensional 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 popular semi-definite programming method is actually a special application (where W (x) is a quadratic function) 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 basic terminologies in continuum physics, such as geometric nonlinearity, internal and external energies, and present "counterexamples" to the Gao-Strang theory based on certain "artificially chosen" operators Λ(x) and quadratic functions F (x). Whereas in the stated contexts, the geometrical operator Λ should be a canonical measure (Cauchy-Reimann type finite deformation operator, see Chapter 6 in [7]) and the external energy F (x) is typically a linear functional on its effective domain; otherwise, its (sub)-differential will not be the external force. Interested readers are refereed by [18] for further discussion.
Remark 3 (Double-Min Duality and Open Problem). The double-min and double-max duality statements were discovered simultaneously in a post-buckling analysis of large deformed beam model [4,5] in 1996, where the finite strain measure Λ is a quadratic differential operator from a 2-D displacement field to a 2-D canonical strain field. Therefore, the triality theory was first proposed in its strong form, i.e. the so-called tri-duality theory (see the next section). Later on when Gao was writing his duality book [7], he realized that this pair of double-min and doublemax dualities holds naturally in convex Hamilton systems. Accordingly, a bi-duality theorem was proposed and proved for geometrically linear systems (where Λ is a linear operator; see Chapter 2 in [7]). Following this, the triality theory was naturally generalized to geometrically nonlinear systems (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-min duality statement needs "certain additional constraints". For the sake of mathematical rigor, the double-min duality was not included in the triality theory and these additional constraints were left as an open problem (see Remark 1 in [9], also Theorem 3 and its Remark in a review article by Gao [10]). By the fact that the double-max duality is always true, the double-min duality was still 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 doublemin 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 triality theory based on linear algebra. To help understanding the intrinsic characteristics of the original problem and its canonical dual, we assume that the nonconvex objective function W (x) is a sum of fourth-order canonical polynomials where B k = B k ij ∈ R n×n , k = 1, · · · , m, are all symmetric matrices, β k > 0 and 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]), which is the mathematical model for natural phenomena of bifurcation and phase transitions in biology, 6 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 canonical function and its Legendre conjugate are quadratic functions, where β = Diag (β k ) represents the diagonal matrix defined by the nonzero vector {β k }.
We will first prove that if n = m, the triality theorem holds in its strong form; otherwise, the theorem holds in its weak form. Three numerical examples are used to illustrate the effectiveness and efficiency of the canonical duality theory.
3. Strong Triality Theory for Quartic Polynomial Optimization: Tri-Duality Theorem. We first consider the case m = n. For simplicity, we assume that β k = 1 in the following discussion (otherwise, B k can be replaced by √ β k B k and d k is replaced by d k / √ β k ). In this case, the problem (1) is denoted as problem (P). Its canonical dual is Theorem 3.1 (Tri-Duality Theorem).
Suppose that m = n, that the assumption (26) is satisfied, thatς is a critical point of Problem (P d ) and thatx = [G (ς)] −1 f . Ifς ∈ S + a , thenς is a global maximizer of Problem (P d ) in S + a if and only ifx is a global minimizer of Problem (P), i.e., the following canonical min-max statement holds: On the other hand, ifς ∈ S − a , then, there exists a neighborhood Xo × So ⊂ R n × S − a of (x,ς), such that either one of the following two statements holds.
(A) The double-min duality statement or (B) the double-max duality statement Proof. Ifς is a critical point of the canonical dual problem (P d ), the criticality condition leads toς = Λ(x). By the fact that To prove the validity of the canonical min-max statement (28), letς be a critical point and ς ∈ S + a . Since Π d (ς) is concave on S + a , the critical pointς ∈ S + a must be a global maximizer of Π d (ς) on S + a . On the other hand, by the convexity of V (ξ), we have Substituting ξ = Λ (x) andξ = Λ (x) into (32), we obtain This leads to By the fact thatς we have For a fixedς ∈ S + a , the convexity of the complementary gap function Gap(x,ς) = 1 2 x, G (ς) x on Xa leads to Therefore, we have This shows thatx is a global minimizer of Problem (P). Since it is assumed thatς ∈ S + a , it follows that (28) is satisfied.
We move on to prove the double-min duality statement (29). Letς be a critical point of Π d (ς) andς ∈ S − a . It is easy to verify that where In light of (31), ∇ 2 Π d (ς) can be expressed in terms ofx = [G (ς)] −1 f as follows: where I is the identity matrix. If the critical pointς ∈ S − a is a local minimizer, we have According to Lemma 6.2 in Appendix, the following matrix inequality is obtained: By the assumption (26),x = [G (ς)] −1 f is also a local minimizer of Problem (P) . Therefore, on a neighborhood Xo × So ⊂ R n × S − a of (x,ς) , we have min Similarly, we can show that ifx is a local minimizer of Problem (P) , the correspondingς is also a local minimizer of Problem (P d ).
The next task is to prove the double-max duality statement (30). Letς ∈ S − a be a local maximizer of Problem (P d ). Then, we have ∇ 2 Π d (ς) 0. This gives us Now we have two possible cases regarding the invertibility of F (x) . If F (x) is invertible, then by using a similar argument as presented above, we can show that the relations hold on a neighborhood Xo × So ⊂ 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 where E T E = I = K T K and D = Diag (σ 1 , · · · , σr, 0, · · · , 0) with σ 1 ≥ σ 2 ≥ · · · ≥ σr > 0 and r = rank (F (x)). Substituting (41) into (40), we obtain Applying Lemma 6.4 in Appendix to (43), it follows that Finally, we have This means thatx is also a local maximizer of Problem (P) under the assumption (26), i.e., there exists a neighborhood X o × So ⊂ R n × S − a of (x,ς) such that max Finally, we can show, in a similar way, that ifx = [G(ς)] −1 f is a local maximizer of Problem (P) andς ∈ S − a , the correspondingς is also a local maximizer of Problem (P d ). Therefore, the tri-duality theorem is proved.
Suppose thatx andς are the critical points of Problem (P) and Problem (P d ), respectively, wherē x = [G (ς)] −1 f . It is easy to verify that To continue, we show the following lemmas.
According to the singular value decomposition theory [22], there exist orthogonal matrices U and E such that Therefore, U is an identity matrix. Let Then, Since ∇ 2 Π d (ς) 0, U is an identity matrix, and E is an orthogonal matrix, we have Thus, a i ≥ λ i , i = 1, · · · , m. Note that Let J = [I m×m , 0 m×(n−m) ] T . Then, we have Let P = TJ. Clearly, rank(P) = m and P T ∇ 2 Π (x) P = Diag (a 1 − λ 1 , · · · , am − λm) 0. The proof is completed.
In a similar way, we can prove the following lemma.
Lemma 4.2. Suppose that m > n. Letx = [G (ς)] −1 f be a critical point, which is a local minimizer of Problem (P), whereς ∈ S − a . Then, there exists a matrix Q ∈ R m×n with rank(Q) = n such that Let p 1 , · · · , pm be the m column vectors of P and let q 1 , · · · , qn be the n column vectors of Q, respectively. Clearly, p 1 , · · · , pm are m independent vectors and q 1 , · · · , qn are n independent vectors. We introduce the following two subspaces Ifς ∈ S + a , then the canonical min-max duality holds in the strong form: Ifς ∈ S − a , then there exists a neighborhood Xo × So ⊂ R n × S − a of (x,ς) such that the double-max duality holds in the strong form However, the double-min duality statement holds conditionally in the following super-symmetrical forms.
From (47), we obtain is a non-singular matrix and rank (F(x)) = m < n. We claim that x = [G (ς)] −1 f is not a local minimizer of Problem (P) . On a contrary, suppose thatx is also a local minimizer. Then, we have Sinceς ∈ S − a and rank (F) = m, it is clear that This is a contradiction. Therefore,x = [G (ς)] −1 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 0 ∈ R m is a local minimizer of the function ϕ(t 1 , · · · , tn).
In light of Lemma 4.1 and the assumption (26), it follows that 0 ∈ R m is, indeed, a local minimizer of the function ϕ(t 1 , · · · , tm).
In a similar way, we can establish the case of m > n. The proof is completed.
Remark 6. Theorem 4.3 shows that both the canonical min-max and double-max duality statements hold strongly for general cases; the double-min duality holds strongly for n = m but weakly for n = m in a symmetrical form. The "certain additional conditions" are simply the intersection Xo X ♭ for m < n and So S ♭ for m > n. Therefore, the open problem left in 2003 [9,10] is solved 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 that it is important to point out that the main results of this (triality) theory are false. This is done by providing elementary counter-examples that lead to think that a correction of this theory is impossible without falling into trivia". It turns out that most of these counter-examples simply use the double-well function W (x) with n = m. In fact, these counter-examples address the same type of 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 for the general case n = m. However, based on the weak double-min duality, one can easily construct other 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 one of their papers. ext Π (x) = 1 2 The canonical dual problem can be expressed as Thus, It is easy to check that Π d (ς) has only one critical pointς 1 = 1 + √ 2, 3.35991198 in S + a and four critical We can verify that Π d (ς) has one critical pointς 1 = −0.90489505 in S + a and two critical pointsς 2 = −0.12552589 andς 3 = −3.974788888 in S − a . Furthermore,ς 2 is a local minimizer andς 3 is a local maximizer of Π d (ς). According to  θp). Then, it is easy to verify that there exists neighborhoods X 0 ⊂ R and S 0 ⊂ R such that 0 ∈ X 0 ,ς 2 ∈ S 0 and min θ∈X 0 This example shows that even if n > m, the canonical min-max duality and the double-max duality 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 = −0.2, B 1 = 0.3, B 2 = 0.7, d 1 = 3, d 2 = 2.7 and f = 1.4. Then, its dual problem is We can verify that Π d (ς) has one critical pointς 1 = (−0.35012607, 3.48303916) T in S + a and two critical pointsς 2 = (−2.98705125, −2.66978626) T andς 3 = (−0.70765026, 2.64881606) T in S − a . Furthermore,ς 2 is a local maximizer andς 3 is a saddle point of Π d (ς). According to Theorem 4.3,x 1 = G(ς 1 ) −1 f = 4.20307342 is the unique global minimizer of Π(x),x 2 = G(ς 2 ) −1 f = −0.29381114 is a local maximizer of Π(x). Let q = (1, 0) T and ψ(ϑ) = Π d (ς 3 + ϑq). Then, it is easy to verify that there exists neighborhoods X 0 ⊂ R and S 0 ⊂ R such thatx 3 ∈ X 0 , 0 ∈ S 0 and min x∈X 0 This example shows that if n < m, the canonical min-max duality and the double-max duality still 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 without input (f = 0) (P 2 ) : ext Π (x) = 1 2 Problem (P 2 ) has four global minimizersx 1 = (1, 0) ,x 2 = (0, −1) ,x 3 = (0, 1) ,x 4 = (−1, 0) and the optimal cost value is 0. Its canonical dual problem is Π d (ς) has only one critical pointς = − 1 2 , − 1 2 ∈ S − a . Furthermore, we can check thatx = [G (ς)] −1 f = (0, 0) is a local maximizer of Problem (P 2 ) . Thus, we cannot use the canonical dual transformation method to obtain the global minimizer of Problem (P 2 ) since this problem is in a perfect symmetrical form without input, which allows more than one global minimizer. Now we perturb Problem (P 2 ) as follows.
In a similar way, we can show that if −D T P −1 D − U −1 0, then P + DUD T 0. The proof is thus completed.
7. Conclusion Remarks. In this paper, we presented a rigorous proof of the double-min duality in the triality theory for a quartic polynomial optimization problem based on elementary linear algebra. Our results show that under some proper assumptions, the triality theory for a class of quartic polynomial optimization problems holds strongly in the tri-duality form if the primal problem and its canonical dual have the same dimension. Otherwise, both the canonical min-max and the double-max still hold strongly, but the double-min duality holds weakly in a symmetric form.