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Journal ArticleDOI

On Fan's minimax theorem

01 Mar 1986-Mathematical Programming (Springer-Verlag)-Vol. 34, Iss: 2, pp 232-234
TL;DR: A new brief proof of Fan's minimax theorem for convex-concave like functions is established using separation arguments.
Abstract: A new brief proof of Fan's minimax theorem for convex-concave like functions is established using separation arguments.
Citations
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Book
25 Jan 2010
TL;DR: Convexity is a natural and powerful property of functions that plays a significant role in many areas of mathematics, both pure and applied as mentioned in this paper, and is an important tool in optimization, mathematical programming and game theory.
Abstract: Like differentiability, convexity is a natural and powerful property of functions that plays a significant role in many areas of mathematics, both pure and applied. It ties together notions from topology, algebra, geometry and analysis, and is an important tool in optimization, mathematical programming and game theory. This book, which is the product of a collaboration of over 15 years, is unique in that it focuses on convex functions themselves, rather than on convex analysis. The authors explore the various classes and their characteristics and applications, treating convex functions in both Euclidean and Banach spaces. The book can either be read sequentially for a graduate course, or dipped into by researchers and practitioners. Each chapter contains a variety of specific examples, and over 600 exercises are included, ranging in difficulty from early graduate to research level.

341 citations

Journal ArticleDOI
TL;DR: In this paper, a subdifferential calculus for lower semicontinuous functions is developed for studying constrained optimization problems, nonclassical problems of calculus of variations, and generalized solutions of first-order partial differential equations on smooth manifolds.
Abstract: We study infinitesimal properties of nonsmooth (nondifferentiable) functions on smooth manifolds. The eigenvalue function of a matrix on the manifold of symmetric matrices gives a natural example of such a nonsmooth function. A subdifferential calculus for lower semicontinuous functions is developed here for studying constrained optimization problems, nonclassical problems of calculus of variations, and generalized solutions of first-order partial differential equations on manifolds. We also establish criteria for monotonicity and invariance of functions and sets with respect to solutions of differential inclusions.

118 citations


Cites methods from "On Fan's minimax theorem"

  • ...(4.16) Since the functional Φ(η, v) − 〈ξ, v〉 is continuous in v in the weak-star topology and is bilinear, M is convex and W is convex and compact, we can apply the nonsymmetric minimax theorem of Borwein-Zhang [15] to obtain from (4.16) that sup η∈M inf v∈W [ Φ(η, v) − 〈ξ, v〉 ] = 0....

    [...]

  • ...Since the functional Φ(η, v) − 〈ξ, v〉 is continuous in v in the weak-star topology and is bilinear, M is convex and W is convex and compact, we can apply the nonsymmetric minimax theorem of Borwein-Zhang [15] to obtain from (4....

    [...]

Book ChapterDOI
01 Jan 1995
TL;DR: A minimax theorem is a theorem which asserts that, under certain conditions, a nonempty set is a minimax set as mentioned in this paper, i.e., a set whose members are nonempty sets.
Abstract: We suppose that X and Y are nonempty sets and f: X x Y →IR A minimax theorem is a theorem which asserts that, under certain conditions, $$\mathop{{\min }}\limits_{Y} \mathop{{\max }}\limits_{X} f = \mathop{{\max }}\limits_{X} \mathop{{\min }}\limits_{Y} f.$$

95 citations

Posted Content
TL;DR: In this paper, the Gibbs measure associated to a logarithmically correlated random potential (including two dimensional free fields) at low temperature is considered and it is shown that the energy landscape freezes and enters in the so-called glassy phase.
Abstract: In this paper, we consider the Gibbs measure associated to a logarithmically correlated random potential (including two dimensional free fields) at low temperature. We prove that the energy landscape freezes and enters in the so-called glassy phase. The limiting Gibbs weights are integrated atomic random measures with random intensity expressed in terms of the critical Gaussian multiplicative chaos. This could be seen as a first rigorous step in the renormalization theory of super-critical Gaussian multiplicative chaos.

54 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that these results form an equivalent chain and this chain includes the strong separation result in finite dimensional spaces between two disjoint closed convex sets of which one is compact.

47 citations

References
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Journal ArticleDOI
TL;DR: Come with us to read a new book that is coming recently, this is a new coming book that many people really want to read will you be one of them?
Abstract: Come with us to read a new book that is coming recently. Yeah, this is a new coming book that many people really want to read will you be one of them? Of course, you should be. It will not make you feel so hard to enjoy your life. Even some people think that reading is a hard to do, you must be sure that you can do it. Hard will be felt when you have no ideas about what kind of book to read. Or sometimes, your reading material is not interesting enough.

1,706 citations

Book
01 Jul 1970

800 citations


"On Fan's minimax theorem" refers background in this paper

  • ...The general form of Fan's minimax theorem [2, 4 ] is:...

    [...]

  • ...f(x, Y3) ~> tf(x, Yl) + (1 - t)f(x, Y2) (2) for all x in X, [ 4 ]....

    [...]

Journal ArticleDOI
TL;DR: In this paper, it was shown that duality gaps can be closed under broad hypotheses in minimax problems, provided certain changes are made in the maximum part of the saddle function which increase its value.
Abstract: We show that duality gaps can be closed under broad hypotheses in minimax problems, provided certain changes are made in the maximum part which increase its value. The primary device is to add a linear perturbation to the saddle function, and send it to zero in the limit. Suprema replace maxima, and infima replace minima. In addition to the usual convexity-concavity type of assumptions on the saddle function and the sets, a form of semireflectivity is required for one of the two spaces of the saddle function. A sharpening of the results is possible when one of the spaces is finite-dimensional. A variant of the proof of the previous results leads to a generalization of a result of Sion, from which the theorem of Kneser and Fan follows.

7 citations