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A NOTE on GENERALIZED INTERIORS of SETS Let X Be A

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A NOTE on GENERALIZED INTERIORS of SETS Let X Be A DIDACTICA MATHEMATICA, Vol. 31(2013), No 1, pp. 37{42 A NOTE ON GENERALIZED INTERIORS OF SETS Anca Grad Abstract. This article familiarizes the reader with generalized interiors of sets, as they are extremely useful when formulating optimality conditions for problems defined on topological vector spaces. Two original results connecting the quasi- relative interior and the quasi interior are included. MSC 2000. 49K99, 90C46 Key words. convex sets, algebraic interior, topological interior, quasi interior, quasi-relative interior 1. INTRODUCTION Let X be a topological vector space and let X∗ be the topological dual space of X. Given a linear continuous functional x∗ 2 X∗ and a point x 2 X, we denote by hx∗; xi the value of x∗ at x. The normal cone associated with a set M ⊆ X is defined by fx∗ 2 X∗ : hx∗; y − xi ≤ 0 for all y 2 Mg if x 2 M N (x) := M ; otherwise. Given a nonempty cone C ⊆ X, its dual cone is the set C+ := fx∗ 2 X∗ : hx∗; xi ≥ 0 for all x 2 Cg: We also use the following subset of C+: (1) C+0 := fx∗ 2 X∗ : hx∗; xi > 0 for all x 2 Cnf0gg: 2. GENERALIZED INTERIORS OF SETS. DEFINITIONS AND BASIC PROPERTIES Interior and generalized interior notions of sets are highly important when tackling optimization problems, as they are often used in formulating regular- ity conditions. Let X be a nontrivial vector space, and let M ⊆ X be a set. The algebraic interior of M is defined by x 2 X : for each y 2 X there exists δ > 0 such that core M := : x + λy 2 M for all λ 2 [0; δ] The intrinsic core of M is defined by x 2 X : for each y 2 aff(M − M) there exists δ > 0 icr M := such that x + λy 2 M for all λ 2 [0; δ] 38 Anca Grad 2 (see Holmes G. B. [11], Zalinescu˘ C. [16]). From the definitions mentioned abobe it follows that core M ⊆ M and core M ⊆ icr M: When M is a convex set, then we have: core M = fx 2 M : cone(M − x) = Xg; and icr M = fx 2 M : cone(M − x) is a linear subspace of Xg: When M is a convex cone, then the set f0g [ core M is also a convex cone and core M = M + core M: Let X be a topological vector space, and let M ⊆ X be a set. We denote by int M the interior of M and by cl M the closure of M. When M is a convex set, both sets int M and cl M are convex. Moreover, if M is convex and int M 6= ;, then we have int M = int(cl M) and cl M = cl(int M): For each set M ⊆ X it holds int M ⊆ core M. The equality int M = core M is true if M is a convex set and one of the following conditions is fulfilled: int M 6= ;; X is a Banach space and M is closed; X is finite dimensional. Consider now a separated topological vector space X, and let M ⊆ X be a set. The strong quasi-relative interior of M is icr M if aff M is closed sqri M := ; otherwise (see Borwein J. M., Jeyakumar V., Lewis A. S. and Wolkowicz H. [2], Jeyakumar V. and Wolkowicz H.[12], Zalinescu˘ C. [16]). When the set M is convex, then sqri M = fx 2 M : cone(M − x) is a closed linear subspace of Xg: The quasi-relative interior of an arbitrary set M ⊆ X is qri M := x 2 M : cl cone(M − x) is a linear subspace of X (see Borwein J. M. and Lewis A. S. [3]). The quasi interior of a set M ⊆ X is tightly connected to the quasi-relative interior and appeared in the literature prior to it. It is defined by qi M := x 2 M : cl cone(M − x) = X : When M is a convex set, then qi M ⊆ qri M and qrifxg = fxg for all x 2 X: Moreover, whenever qi M 6= ;, then qi M = qri M. The following chain of inclusions holds for an arbitrary set M ⊆ X: core M ⊆ sqri M ⊆ icr M: 3 A Note on Generalized Interiors of Sets 39 When X is a separated locally convex space, and M ⊆ X is a convex set, then the following chains of inclusions hold: (2) int M ⊆ core M ⊆ sqri M ⊆ icr M ⊆ qri M ⊆ M; (3) int M ⊆ core M ⊆ qi M ⊆ qri M ⊆ M: When int M 6= ;, then all the generalized interior notions in (2) and (3) collapse in equality to int M. n n Let us consider the case when X = R , with n 2 N, and let M ⊆ R be a set. Then the relative interior of M is defined by ri M := fx 2 aff M : there exists " > 0 such that B(x; ") \ aff M ⊆ Mg ; where B(x; ") is the closed ball centered at x with radius ", in the Euclidian norm. In this finite dimensional setting, when M is a convex set, then the following equalities hold: (4) qri M = sqri M = icr M = ri M (according to Borwein J. M. and Lewis A. S. [3], Gowda M. S. and Teboulle M. [8]). As well, the following chain of equalities proves to be valid: (5) core M = qi M = int M (according to Limber M.A. and Goodrich R.K. [13], Rockafellar R. T. [14]). Let now X be a separated locally convex space, and let C ⊆ X be a nonempty closed convex cone. Then qi(C+) = C+0 (see for example Bot¸R. I., Grad S. M. and Wanka G. [6, Proposition 2.1.1]). The equality above is mainly the reason why the set C+0 defined by (1) is called the quasi-relative interior of the dual cone of C (even in the more general case when C is not closed). We refer the reader to Bot¸R. I. and Csetnek E. R. [4] for a recent synthesis with respect to regularity conditions by means of generalized interiors along with new achievements in the same area. We continue by presenting some characterizations of the quasi interior and quasi-relative interior of convex sets in separated locally convex spaces. Theorem 2.1 (Borwein J. M., Lewis A. S.[3]). Let M be a convex subset of a separated locally convex space X, and let x 2 M. Then ∗ x 2 qri M if and only if NM (x) is a linear subspace of X : The following characterization of the quasi interior of a convex set was extended to separated locally convex spaces by Bot¸R. I., Csetnek E. R. and Wanka G. [5]. 40 Anca Grad 4 Theorem 2.2. Let M be a convex subset of a separated locally convex space X, and let x 2 M. Then x 2 qi M if and only if NM (x) = f0g: Some useful properties of the quasi-relative interior of a convex set are listed below. For their proofs we refer the reader to Borwein J. M. and Goebel R.[1], Borwein J. M. and Lewis A. S.[3], and Bot¸R. I., Csetnek E. R. and Wanka G. [5]. Proposition 2.3. Let M and N be convex subsets of a separated locally convex space X. Then the following statements are true: (i) qri M + qri N ⊆ qri(M + N); (ii) qri M × qri N = qri(M × N); (iii) qri(M − x) = (qri M) − x for all x 2 X; (iv) qri(αM) = α qri M for all α 2 Rnf0g; (v) λ qri M +(1−λ)M ⊆ qri M for all λ 2 (0; 1], whence qri M is a convex set; (vi) qri(qri M) = qri M; (vii) if M is affine, then qri M = M; (viii) if qri M 6= ;, then cl qri M = cl M and cl cone qri M = cl cone M. Remark 2.4. For two convex subsets M and N of a separated locally convex X such that M ⊆ N, it holds (6) qi M ⊆ qi N; a property which is no longer true for the quasi-relative interior. However, according to Cammaroto F. and Di Bella B. [7, Proposition 1.12], when aff M = aff N, then qri M ⊆ qri N holds. Remark 2.5. Let X be a separated locally convex space, and let C ⊆ X be a convex cone. Then, using Proposition 2:3 (v) and Remark 2:4 we deduce that the equality (7) qi C + C = qi C holds. 3. A CONNECTION BETWEEN THE QUASI INTERIOR AND THE QUASI-RELATIVE INTERIOR OF A CONVEX SET In this section we present two original results of the author, the first of them linking in a new way the quasi interior and the quasi-relative interior of a convex set. 5 A Note on Generalized Interiors of Sets 41 Theorem 3.1 (Grad A.). Let M be a convex subset of a separated locally convex space X, and let x 2 M: Then 0 2 qi(M − M) x 2 qi M () x 2 qri M: Proof. Necessity. From x 2 qi M and Theorem 2.2 it follows that NM (x) = f0g. On the other hand, since qri M = qi M, we obviously have x 2 qri M. ∗ Let now x 2 NM−M (0). This means that hx∗; y − zi ≤ 0 for all y; z 2 M: By taking z := x, we get that hx∗; y − xi ≤ 0 for all y 2 M; ∗ ∗ which means that x 2 NM (x).
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