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Dimension in Polynomial Variational Inequalities DIMENSION IN POLYNOMIAL VARIATIONAL INEQUALITIES VU TRUNG HIEU∗ Abstract. The aim of the paper is twofold. Firstly, by using the constant rank level set theorem from differential geometry, we establish sharp upper bounds for the dimensions of the solution sets of polynomial variational inequalities under mild conditions. Secondly, a classification of polynomial variational inequalities based on dimensions of their solution sets is introduced and investigated. Several illustrative examples are provided. Key words. polynomial variational inequality, polynomial fractional optimization, semialge- braic set, solution set, dimension AMS subject classifications. 90C33, 14P10 1. Introduction. We consider the following variational inequality find x 2 K such that hF (x); y − xi ≥ 0 8y 2 K; n n n where K is a semialgebraic set in R and F : R ! R is a polynomial map. The problem and its solution set are denoted by PVI(K; F ) and Sol(K; F ), respectively. This problem is an natural extension of the well-known linear complementarity prob- lem, the linear variational inequality, the tensor complementarity problem, and the polynomial complementarity problem (see, e.g., [2,5,7,8]) which have received a lot of attention from researchers. Thanks to the Tarski-Seidenberg theorem [3, Theorem 2.6], Sol(K; F ) set is semi- algebraic, hence that it has finitely many connected components and each component is path-connected. Furthermore, the dimension of Sol(K; F ) is well-defined. The present paper focuses on this topic. Firstly, we show a sharp upper bound for dim(Sol(K; F )) provided that the prob- lem satisfies the Abadie constraint qualification and the constant rank condition. Re- sults for the finiteness of Sol(K; F ) are obtained. Secondly, we show that stationary points of the polynomial fractional optimization problem [10] p(x) minimize subject to x 2 K; q(x) where p(x) and q(x) are polynomials, q(x) > 0 for all x 2 K, is the solution set of a certainly polynomial variational inequality. Hence, an estimate for the dimension of these stationary points is established. Thirdly, based on dimensions of the solution sets, a classification of polynomial variational inequalities is introduced. We also arXiv:2001.09435v1 [math.OC] 26 Jan 2020 discuss thickness of the classes. The organization of the paper is as follows. Section2 gives a brief introduction to semialgebraic geometry and polynomial variational inequalities. Section3 shows up- per bounds for the dimension of solution sets. Finiteness of solution sets is discussed in Section4. Section5 investigates stationary points in polynomial fractional opti- mization. A classification of polynomial variational inequalities is studied in Section 6. The last section gives some concluding remarks. 2. Preliminaries. In this section, we will recall some definitions, notations, and auxiliary results from semialgebraic geometry and polynomial variational inequalities. ∗LIP6, Sorbonne University, Paris, France ([email protected], [email protected]). 1 2 VU TRUNG HIEU n 2.1. Semialgebraic Sets. Recall a subset in R is semialgebraic [1, Definition 2.1.4], if it is the union of finitely many subsets of the form n x 2 R : f1(x) = ··· = f`(x) = 0; g`+1(x) < 0; : : : ; gm(x) < 0 ; where `; m are natural numbers, and f1; : : : ; f`; g`+1; : : : ; gm are polynomials with real coefficients. The semialgebraic property is preserved by taking finitely union, intersection, minus and taking closure of semialgebraic sets. The well-known Tarski-Seidenberg theorem [3, Theorem 2.3] states that the image of a semialgebraic set under a linear projection is a semialgebraic set. m n Let S1 ⊂ R and S2 ⊂ R be semialgebraic sets. A vector-valued map G : S1 ! S2 is said to be semialgebraic [1, Definition 2.2.5], if its graph gph(G) = f(x; v) 2 S1 × S2 : v = G(x)g m n is a semialgebraic subset in R × R . m Let S be a semialgebraic set of R . Then there exists a decomposition of S into a disjoint union of semialgebraic subsets [1, Theorem 2.3.6] S = S1 [···[ Ss; di 0 where each Si is semialgebraically diffeomorphic to (0; 1) , i 2 [s]. Here, let (0; 1) di di be a point, (0; 1) ⊂ R is the set of points x = (x1; : : : ; xdi ) such that xj 2 (0; 1) for all j = 1; : : : ; di. The dimension of S is, by definition [3, Proposition 3.15], dim(S) := maxfd1; :::; dsg: The dimension is well-defined and not depends on the decomposition of S. We adopt the convention that dim(;) := −∞. If S is nonempty and dim(S) = 0 then S has finitely many points. m n Assume that S ⊂ R is a semialgebraic set and G : S ! R is a semialgebraic map. Theorem 3.18 in [3] says that dim(G(S)) ≤ dim(S). Let S1; :::; Sk be semialge- n braic sets in R . Applying [1, Proposition 2.8.5], one has the following equality: (2.1) dim(S1 [···[ Sk) = maxfdim S1;:::; dim Skg: m Let S1;S2 are semialgebraic manifolds in R and G : S1 ! S2 be a smooth semialgebraic map. Assume that the rank of the Jacobian of G is k in a neighborhood −1 of the level set G (v), where v 2 S2 be given. The constant rank level set theorem −1 −1 [9, Theorem 11.2] says that G (v) is a submanifold of S1 and dim(G (v)) = m − k. 2.2. Polynomial Variational Inequalities. Let K be a nonempty semialge- n n n braic closed convex subset in R and F : R ! R be a polynomial map. The polynomial variational inequality defined by K and F is the following problem: find x 2 K such that hF (x); y − xi ≥ 0 8y 2 K; n where hx; yi is the usual scalar product of x; y in the Euclidean space R . We will respectively write the problem and its solution set PVI(K; F ) and Sol(K; F ). Note that x solves (PVI) if and only if F (x) 2 −NK (x); where NK (x) is the normal cone of K at x 2 K which is defined by ∗ n ∗ NK (x) = fx 2 R : hx ; y − xi ≤ 0 8y 2 Kg: DIMENSION IN POLYNOMIAL VARIATIONAL INEQUALITIES 3 Clearly, if x belongs to the interior of K, then x 2 Sol(K; F ) if and only if F (x) = 0. n When K = R , x solves PVI(K; F ) if and only if x is a zero point of the function F . Throughout the work, we assume that K given by finitely many convex polynomial functions gi(x), i 2 [m] := f1; : : : ; mg; and finitely many affine functions hj(x); j 2 [`]; as follows n K = x 2 R : gi(x) ≤ 0; i 2 [m]; hj(x) = 0; j 2 [`] : To find the solution set of PVI(K; F ), we will find the solutions on each pseudo- face of K. For every index set α ⊂ [m], we associate that with the following pseudo- face n Kα = fx 2 R : gi(x) = 0; 8i 2 α; gi(x) < 0; 8i 2 [m] n α; hj(x) = 0; 8j 2 [`]g : All pseudo-faces establish a finite disjoint decomposition of K. Therefore, we have [ (2.2) K = Kα: α⊂[m] Since Sol(K; F ) is a subset of K, from the disjoint decomposition (2.2) we have the following equality: [ (2.3) Sol(K; F ) = [Sol(K; F ) \ Kα] α⊂[m] By applying the Tarski-Seidenberg theorem with quantifiers [3, Theorem 2.6], we see n that the solution set Sol(K; F ) is semialgebraic in R . From (2.3) and (2.1), one concludes that (2.4) dim(Sol(K; F )) = max fdim(Sol(K; F ) \ Kα)g: α⊂[m] The Bouligand-Severi tangent cone (see, e.g., [4, p. 15]) of K at x 2 K, denoted n by TK (x), consists of the vectors v 2 R , called the tangent vectors to K at x, for which there exist a sequence of vectors fykg ⊂ K and a sequence of positive scalars ftkg such that yk − x lim yk = x; lim tk = 0; and lim = v: k!1 k!1 k!1 tk The linearization cone (see, e.g., [4, p. 17]) of K at x is defined by n LK (x) = v 2 R : hrgi(x); vi ≤ 0; i 2 I(x); hrhj(x); vi = 0; j 2 [`] with I(x) := fi 2 [m]: gi(x) = 0g denoting the active index set at x. One says that K satifies the Abadie constraint qualification (ACQ) [4, p. 17] at x 2 K if LK (x) = TK (x). If the ACQ holds at every point of K, then the Karush-Kuhn- Tucker conditions can be applied to variational inequalities with the constraint set K m ` [4, Proposition 1.3.4], i.e., x 2 Sol(K; F ) if and only if there exist λ 2 R and µ 2 R such that X X F (x) + λirgi(x) + µirhj(x) = 0; (2.5) i2[m] j2[`] λT g(x) = 0; λ ≥ 0; g(x) ≤ 0; h(x) = 0: 4 VU TRUNG HIEU One says that K satisfies the linearly independent constraint qualification, is writ- ten LICQ for brevity, if the gradient vectors frgi(x); rhj(x); i 2 I(x); j 2 [`]g are linearly independent, for all point x 2 K. If the LICQ holds on K, then the ACQ (see, e.g. [4, p. 17]) also holds on K. 3. Sharp Upper Bounds for Dimensions. This section gives sharp upper bounds for the dimensions of the solution sets of polynomial variational inequalities provided that the problems satisfy the constant rank condition. Consequently, some special cases are considered.
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