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Well-Positioned Convex Sets and Functions and Applications Michel Thera´ LACO, UMR-CNRS 6090, University of Limoges [email protected]

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Well-Positioned Convex Sets and Functions and Applications Michel Thera´ LACO, UMR-CNRS 6090, University of Limoges Michel.Thera@Unilim.Fr Well-positioned convex sets and functions and applications Michel Thera´ LACO, UMR-CNRS 6090, University of Limoges [email protected] Main collaborators : S. Adly, E. Ernst and C. Zalinescu Naples, december 19 2005 p.1/?? Credits E. ERNST and M. THERA´ ,C.ZALINESCU˘ , Constrained optimization and strict convex separation, Slice-continuous sets in reflexive Banach spaces: convex constrained optimization and strict convex separation, Journal of Functional Analysis, Vol 223/1 pp 179-203, 2005. S. ADLY,E.ERNST and M. THERA´ , Well-positioned closed convex sets and well-positioned closed convex functions, Journal of Global Optimization 29 (4): 337-351, (2004). E. ERNST and M. THERA´ , Continuous sets and non-attaining functionals in reflexive Banach spaces, Variational Analysis and Applications, F. Giannessi and A. Maugeri, Eds, Springer, 2005 in press. p.2/?? Credits S. ADLY,E.ERNST and M. THERA´ , Stability of Non-coercive Variational Inequalities, Communications in Contemporary Mathematics, Vol 4, 1, 145 – 160, 2002. S. ADLY,E.ERNST and M. THERA´ , On the converse of the Dieudonné Theorem in Reflexive Banach Spaces, Special volume, dedicated to the memory of Boris Pshenichnyi, Cybernetics & System Analysis, No 3, 34 – 39, 2002. S. ADLY,E.ERNST and M. THERA´ , On the closedness of the algebraic difference of closed convex sets, Journal de Mathématiques Pures et Appliquées, Volume 82, No 9, 1219 – 1249, 2003. p.3/?? Outline Setting p.4/?? Outline Setting Some properties of the barrier cone to a convex set p.5/?? Outline Setting Some properties of the barrier cone to a convex set Well-positioned sets p.6/?? Outline Setting Some properties of the barrier cone to a convex set Well-positioned sets Equivalent characterizations of W-P p.7/?? Outline Setting Some properties of the barrier cone to a convex set Well-positioned sets Equivalent characterizations of W-P Well-positioned convex functionals p.8/?? Outline Setting Some properties of the barrier cone to a convex set Well-positioned sets Equivalent characterizations of W-P Well-positioned convex functionals Stability of the solution set of a semi-coercive VI p.9/?? Outline Setting Some properties of the barrier cone to a convex set Well-positioned sets Equivalent characterizations of W-P Well-positioned convex functionals Stability of the solution set of a semi-coercive VI Closure of the difference of closed convex sets p.10/?? Outline Setting Some properties of the barrier cone to a convex set Well-positioned sets Equivalent characterizations of W-P Well-positioned convex functionals Stability of the solution set of a semi-coercive VI Closure of the difference of closed convex sets Continuous and Slice-continuous sets p.11/?? Outline Setting Some properties of the barrier cone to a convex set Well-positioned sets Equivalent characterizations of W-P Well-positioned convex functionals Stability of the solution set of a semi-coercive VI Closure of the difference of closed convex sets Continuous and Slice-continuous sets Separation of convex sets p.12/?? Framework Throughout this presentation, we suppose that X is a re- flexive Banach space (unless otherwise stated) with con- tinuous dual X∗. p.13/?? An important object: the barrier cone The recession cone to the closed convex set S is the closed convex cone S∞ defined as S∞ = {v ∈ X : ∀ λ > 0, ∀ x0 ∈ S, x0 + λv ∈ S} . S is called linearly bounded whenever S∞ = {0}. Given a closed convex subset S of X, the domain of the support function given by σS(f) := suphf,xi x∈S is the barrier cone of S: ∗ B(S) = {f ∈ X : σS(f) < +∞} = Dom σS. p.14/?? The closure of the barrier cone Let C be a closed convex set in a normed linear space X. When X is reflexive: By the Bipolar Theorem, B(C) = B(C)◦◦ But ◦◦ ◦ B(C) =(C∞) . Thus, B(C) is characterized in X∗ by the formula: B(C)=(C∞)◦. p.15/?? Setting: Normed linear space Set Cf = {x ∈ X : hf,xi ≥ kxk} . For every closed convex set C of a normed linear space X, the following two facts are equivalent: B(C)=(C∞)◦; ∞ ◦ C ∩ Cf is bounded for every f ∈ (C ) . ∗ When C is linearly bounded B(C) = X if and only if C ∩ Cf is bounded for every f ∈ X∗. The second condition defines the family of conically bounded sets which is a subclass of the class of linearly bounded sets for which holds the property of density of the barrier cone in X∗. p.16/?? The closure of the barrier cone We define the temperate cone of C: kxk T (C) = f ∈ X∗ : lim inf = ∞ . r→∞ x∈C,hf,xi≥r r As infx∈∅ kxk = ∞, it follows that B(C) ⊆T (C). The temperate cone of a closed convex set is norm-closed. FACT [AET, PAMS, 2003] : For every closed convex set C of a normed linear space X, the closure of the barrier cone is the temperate cone. B(C) = T (C). p.17/?? Closure of the domain of the Fenchel conjugate of Ψ ∈ Γ0(X) Take 2 2 k(x,µ)kX×R := kxk + µ , ∀(x,µ) ∈ X × R, p as norm on X × R and ∗ 2 2 ∗ R k(f, λ)kX ×R := kfk∗ + λ , ∀(f, λ) ∈ X × p as dual norm on X∗ × R. The duality pairing is given ∀(f, λ) ∈ X∗ × R, (x,µ) ∈ X × R by h(f, λ), (x,µ)iX∗×R,X×R = hf,xi + λµ p.18/?? Closure of the domain of the Fenchel conjugate of Ψ ∈ Γ0(X) Given an extended-real valued function Ψ : X → R ∪ {∞}, recall that the Fenchel conjugate of Ψ is the function Ψ∗ : X∗ → R ∪ {+∞} given by Ψ∗(f) := sup{hf,xi− Ψ(x)}. x∈X Obviously, the domain of Ψ∗ is connected to the barrier cone B(epi Ψ) of the epigraph of Ψ through the following equivalence: g ∈ DomΨ∗ ⇐⇒ (g, −1) ∈B(epi Ψ). This yields DomΨ∗ × {−1} = B(epi Ψ) ∩ X∗ × {−1} . p.19/?? Standard techniques from convex analysis allow us to prove that DomΨ∗ = B(epi Ψ) ∩ (X∗ × {−1}). Precedent results allow us to find a new proof of the well-known characterization of DomΨ∗ Let Ψ : X → R ∪ {+∞} be an extended-real valued proper convex and lower semicontinuous functional. Then 0 ∈ Dom(Ψ∗) ⇐⇒ the map x 7→ (Ψ(x) + εkxk) is bounded below for every ε > 0. p.20/?? Well-Positioned Sets [AET, JOGO, 2004] The concept of well-positioned closed convex set is a geometric notion equivalent, in the framework of reflexive Banach spaces, to the absence of lines and to weak local compactness. The necessity of well-positionedness in this separation problem was established by Adly et al., while sufficiency goes back to Dieudonné A nonempty subset C of the normed vector space X is ∗ well-positioned if there exist x0 ∈ X and g ∈ X such that: hg,x − x0i ≥ kx − x0k, ∀ x ∈ C. p.21/?? Equivalently, denoting by Cg := {x ∈ X : hg,xi ≥ kxk}, then C is well-positioned ⇐⇒ C is included in some translate of Cg. g g 2 1 C x 0 C g C g 2 1 p.22/?? A closed convex set C is well-positioned ⇐⇒ for every y ∈ C C there exists R > 0 such that 0 ∈/ co(My,R), where x − y M C = : x ∈ C, |x − y|≥ R . y,R |x − y| p.23/?? C y C R M y,R p.24/?? Analytical characterization of W.P A nonempty closed convex set C of a reflexive Banach space X is well-positioned ⇐⇒ the following two assumptions are satisfied: C contains no lines; xn n n N n , such that 6 ∃{x } ∈ ⊂ C, kx k→ +∞ kxnk * 0 When X is finite dimensional, a nonempty closed convex set is well-positioned ⇐⇒ C∞ is pointed, i.e., C∞ ∩−C∞ = {0}. In particular, every compact and convex set is well-positioned in Rn. p.25/?? A characterization Let C be a nonempty subset of a reflexive Banach space X. The following two conditions are equivalent: The barrier cone of C has a nonempty interior; C is well-positioned. Moreover, if Int B(C) =6 ∅, then, Int B(C)=Int(C∞)◦. p.26/?? Well-positioned functionals We say that a proper convex lower semicontinuous functional Ψ : X → R ∪ {+∞} is well-positioned if the epigraph of Ψ, epi Ψ = {(x, λ) ∈ X × R : λ ≥ Ψ(x)}, is a well-positioned subset of X × R. Let Ψ be a proper lower semicontinuous convex function on a reflexive Banach space. Then, Int DomΨ∗ =6 ∅ ⇐⇒ Ψ is well − positioned. p.27/?? Well-positioned functionals Ψ ∈ Γ0(X) is well-positioned ⇐⇒ if the two following assumptions hold: Ker (Ψ∞) contains no lines; xn N , such that and 6∃(xn)n∈ ⊂ DomΨ, kxnk→ +∞ kxnk * 0 Ψ(xn) . kxnk → 0 We also racapture the following result : g ∈ Int DomΨ∗ if and only if the functional Ψ − g is coercive, i.e., Ψ(x) − hg,xi lim inf > 0. kxk→+∞ kxk p.28/?? Stability of the existence of the solution for semi-coercive varia- tional inequalities Framework : V I(A,f, Φ, K) : find u ∈ K ∩ DomΦ such that hAu − f,v − ui + Φ(v) − Φ(u) ≥ 0, ∀v ∈ K, (1) Assumptions : K is a closed convex set in a reflexive Banach space X f ∈ X∗ Φ ∈ Γ0(X) is assumed to be bounded below K ∩ DomΦ =6 ∅ A is a semi-coercive operator from X to X∗ A is pseudomonotone in the sense of Brezis. p.29/?? Stability of the existence of the solution for semi-coercive varia- tional inequalities Recall that A is semicoercive if 2 hAv − Au,v − ui≥ κ (distU (v − u)) ∀u, v ∈ X (2) A(x + u) = A(x) ∀x ∈ X and u ∈ U, and A(X) ⊆ U ⊥, for some positive constant κ and some closed subspace U of X.
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