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Friday, August 7, 2020 | History

2 edition of On sufficient conditions for the sum of two weak * closed convex sets to be weak * closed found in the catalog.

# On sufficient conditions for the sum of two weak * closed convex sets to be weak * closed

## by M. Ali Khan

Written in English

Edition Notes

Includes bibliographical references (p. 5).

The Physical Object ID Numbers Statement M. Ali Khan, Rajiv Vohra Series BEBR faculty working paper -- no. 85-1202, BEBR faculty working paper -- no. 85-1202. Contributions Vohra, Rajiv, University of Illinois at Urbana-Champaign. College of Commerce and Business Administration Pagination 5 p. ; Open Library OL25113469M OCLC/WorldCa 741956446

3 Prove that the intersection of two convex sets is a convex set. Proof: Let A and B be convex sets. We want to show that A ∩ B is also convex. Take x1,x2 ∈ A ∩ B, and let x lie on the line segment between these two points. Then x ∈ A because A is convex, and similarly, x ∈ B because B is convex. Therefore x ∈ A ∩ B, as desired. positive weighted sum & composition with aﬃne function f is convex =⇒ αf is convex for any α ≥ 0 f1,f2 are convex =⇒ f1 +f2 is convex (extends to inﬁnite sums, integrals) f is convex =⇒ f(Ax +b) is convex Examples: log barrier for linear inequalities f (x) = − Xm i=1 log(bi −aT i x) f (x) = kAx +bk.

{ convex functions are exactly the functions with convex epigraphs. Convexity of level sets speci es a wider family of functions, the so called quasiconvex ones. Proposition + [Closeness of level sets] If a convex function f is closed, then all its level sets are closed. Recall that an empty set is closed (and, by the way, is open. Indifference curves can be either convex or strictly convex, but interior solutions generally only happen when they are strictly convex. Presence of a tangent point (between a budget constraint and indifference curve) is a sufficient condition for strict convexity of indifference curves.

De nition: A set S in a vector space V is convex if for any two points xand yin S, and any in the unit interval [0;1], the point (1)x+ yis in S. Theorem: The intersection of any collection of convex sets is convex | i.e., if for each in some set Athe set S is convex, then the set T 2A S is convex. The primary aim of this book is to present the conjugate and subdifferential calculus using the method of perturbation functions in order to obtain the most general results in this field. The secondary aim is to provide important applications of this calculus and of the properties of convex functions.

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### On sufficient conditions for the sum of two weak * closed convex sets to be weak * closed by M. Ali Khan Download PDF EPUB FB2

10COP STX BEBR FACULTYWORKING PAPERNO ob OnSufficientConditionsfortheSumofTwo Weak*ClosedConvexSetstobeWeak*Closed. Cite this article. Ali Khan, M., Vohra, R. On sufficient conditions for the sum of two weak closed convex sets to be weak closed.

by: 5. On sufficient conditions for the sum of two weak * closed convex sets to be weak * closed / By M. Ali. Khan, Rajiv. Vohra and University of Illinois at Urbana-Champaign. College of Commerce and Business Administration. Abstract. Includes bibliographical references (p.

5).Mode of access: Internet. We provide characteristic properties of this class of sets and we relate it to strongly and weakly convex sets via the epigraph and the level sets.

Finally, we give three applications: a separation theorem, a sufficient condition for global optimum of a nonconvex programming problem, and a sufficient geometrical condition for a set to be a Cited by: \$\begingroup\$ As Michael Greinecker says, the closed convex sets differ in general.

We could also consider the kernel of a continuous linear functional. Weak-* closed convex and closed convex are the same if and only if the space is reflexive, i.e. the weak and weak-* topologies coincide. \$\endgroup\$ – Robert Furber Aug 31 '16 at The graph of ∂ MR f is not strong × weak * -closed in general: see, e.g., [8] for a discussion on what would be sufficient to add to the strong × weak * topology on X × X * to guarantee the.

The following theorem gives us a sufficient condition for the sum of discretely convex sets in Z 2 to be discretely convex. The proof is given in Appendix B.

Theorem 3. Let S 1 and S 2 be nonempty discretely convex subsets of Z 2. If S i + Z + 2 = S i for every i ∈ {1, 2}, then S 1 + S 2 is discretely convex. Convex, concave, strictly convex, and strongly convex functions First and second order characterizations of convex functions Optimality conditions for convex problems 1 Theory of convex functions De nition Let’s rst recall the de nition of a convex function.

De nition 1. A function f: Rn!Ris convex if its domain is a convex set and for. A set C in a real or complex vector space is said to be absolutely convex or disked if it is convex and balanced (some people use the term "circled" instead of "balanced"), in which case it is called a disked hull or the absolute convex hull of a set is the intersection of all disks containing that set.

Lecture 2 Open Set and Interior Let X ⊆ Rn be a nonempty set Def. The set X is open if for every x ∈ X there is an open ball B(x,r) that entirely lies in the set X, i.e., for each x ∈ X there is r > 0 for all z with kz − xk.

Then, using the distance function, in [10, 14] a condition similar to () was shown to be necessary and sufficient for the invariance of closed convex sets, while in [11] a sufficient condition. We present several equivalent conditions for the Karush–Kuhn–Tucker conditions for weak ⁎ compact convex sets.

Using them, we extend several existing theorems of the alternative in terms of weak ⁎ compact convex sets. Such extensions allow us to express the KKT conditions and hence necessary optimality conditions for more general nonsmooth optimization problems with.

Dual of intersection of cones. Let Cand Dbe closed convex cones in Rn. In this problem we will show that (C\D) = C + D: Here, + denotes set addition: C + D is the set fu+ vju2C; v2Dg. In other words, the dual of the intersection of two closed convex cones is the sum of the dual cones.

The problem is asking to show that closed in the weak topology implies sequentially closed in the weak topology. A solution can be given that does not use the specifics of the weak topology at all.

For any topological space, closed implies sequentially closed. Let be complete and be nonempty, closed sets such that is nonempty. Let be a weakly contractive mapping such that. Assume the pair has the -property. Then, there exists unique in such that. The following theorem tells that the sum of two non-self-operators has the.

We also show that the strong CHIP is weaker than any of the weak Slater conditions that one can naturally impose on the sets in question. These results generalize the main results of our paper [J. Approx. Theory, 90, pp. ] and hence those of several other papers as well. This chapter is concerned with basic principles of convex programming in Banach spaces, that is, with the minimization of lower-semicontinuous convex functions on closed convex sets.

This is a preview of subscription content, log in to check access. Proof. It is obvious that K is convex and closed (hence by Mazur's theorem weakly closed) in L^i, X). By [3], there exists a reflexive Banach space Y and a one-to-one continuous linear operator T: Y-* X such that K is the image under T of some weakly compact convex set /cK Note that T is weakly continuous; hence T\j is a weak homeomorphism.

Next. Stack Exchange network consists of Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share. Sufficient condition such that weak and initial topology coincide for a locally convex space This is the opposite question to this one: Example of locally convex space such that its weak.

the convex level sets of the functional //* (see Corollaries 2A and 2B). The sufficient condition for weak compactness which is obtained in this way may be regarded as a generalization of Nagumo's Theo-rem [12] in the calculus of variations, and it is also related to recent work of Olech [13, 14] in the same area.

Furthermore, it generalizes.Minkowski sums of convex sets. The Minkowski sum of two compact convex sets is compact. The sum of a compact convex set and a closed convex set is closed.

The following famous theorem, proved by Dieudonné ingives a sufficient condition for the difference of two closed convex subsets to be closed.In this paper we study sufficient conditions for metric subregularity of a set-valued map which is the sum of a single-valued continuous map and a locally closed subset.

First we derive a sufficient condition for metric subregularity which is weaker than the so-called first-order sufficient condition for metric subregularity (FOSCMS) by adding.