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Well-behaved asymptotical convex-concave functions

NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION 1997年第9-10期18卷 1013-1021页

作者： Moudafi, A Univ Limoges LACO URA CNRS 1586 F-87060 Limoges France

Recently Auslender & Cronzeix introduced a well-behaved asymptotical notion for convex functions. The aim of this work is to extend this notion to the saddle case.

关键词： convex-concave functions saddle-values approximate subdifferential epi/hypo-sum well-behaved functions marginal functions proximal algorithm

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SADDLE-POINT DYNAMICS: CONDITIONS FOR ASYMPTOTIC STABILITY OF SADDLE POINTS

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 2017年第1期55卷 486-511页

作者： Cherukuri, Ashish Gharesifard, Bahman Cortes, Jorge Univ Calif San Diego Dept Mech & Aerosp Engn La Jolla CA 92093 USA Queens Univ Dept Math & Stat Kingston ON K7L 3N6 Canada

This paper considers continuously differentiable functions of two vector variables that have (possibly a continuum of) min-max saddle points. We study the asymptotic convergence properties of the associated saddle-point dynamics (gradient descent in the first variable and gradient ascent in the second one). We identify a suite of complementary conditions under which the set of saddle points is asymptotically stable under the saddle-point dynamics. Our first set of results is based on the convexity-concavity of the function defining the saddle-point dynamics to establish the convergence guarantees. For functions that do not enjoy this feature, our second set of results relies on properties of the linearization of the dynamics, the function along the proximal normals to the saddle set, and the linearity of the function in one variable. We also provide global versions of the asymptotic convergence results. Various examples illustrate our discussion.

关键词： saddle-point dynamics asymptotic convergence convex-concave functions proximal calculus center manifold theory nonsmooth dynamics

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Stability and robustness for saddle-point dynamics through monotone mappings

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SYSTEMS & CONTROL LETTERS 2017年 108卷 16-22页

作者： Goebel, Rafal Loyola Univ Chicago Dept Math & Stat 1032 W Sheridan Rd Chicago IL 60660 USA

Motivated by recent interest in saddle-point dynamics, where, given a convex in x and concave in y function, trajectories follow the steepest descent in x and the steepest ascent in y, and where convergence of trajectories to saddle points is desired, this note revisits the maximal monotone mapping approach to saddle-point dynamics, mildly improves one convergence result, and proposes new results on the robustness of pointwise asymptotic stability of the set of saddle points. The results apply to nonsmooth convex/concave functions and constraints, and - as a special case - to projected saddle-point dynamics. (C) 2017 Elsevier B.V. All rights reserved.

关键词： Saddle-point dynamics Monotone mappings convex-concave functions Pointwise asymptotic stability Robustness

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convexity in zero-sum differential games

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 2002年第5期40卷 1491-1504页

作者： Goebel, R Univ British Columbia Dept Math Vancouver BC V6T 1Z2 Canada Simon Fraser Univ Ctr Expt & Construct Math Burnaby BC V5A 1S6 Canada

A new approach to two-player zero-sum differential games with convex-concave cost function is presented. It employs the tools of convex and variational analysis. A necessary and sufficient condition on controls to be an open-loop saddle point of the game is given. Explicit formulas for saddle controls are derived in terms of the subdifferential of the function conjugate to the cost. Existence of saddle controls is concluded under very general assumptions, not requiring the compactness of control sets. A Hamiltonian inclusion, new to the field of differential games, is shown to describe equilibrium trajectories of the game.

关键词： two-player zero-sum differential games open-loop controls convex-concave functions generalized Hamiltonian systems

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Variational inequalities, system of functional equations, and incomplete information repeated games

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 2001年第2期40卷 516-524页

作者： Laraki, R Ecole Polytech Lab Econometrie F-75005 Paris France Univ Paris 10 UFR SEGMI Modal X F-92001 Nanterre 10 France

We consider a pair of functional equations obtained by Mertens and Zamir (Internat. J. Game Theory, 1 (1971-72), pp. 39-64;J. Math. Anal. Appl. 60 (1977), pp. 550-558) to characterize the asymptotic value of a two person zero sum repeated with lack of information on both sides (Aumann and Maschler (Repeated Games with Incomplete Information, MIT Press, Cambridge, MA 1995)). We give a new proof for the convergence of the discounted values of the repeated game and a new characterization of the limit using variational inequalities. The same idea allows us to prove existence and uniqueness of a Lipschitz solution for the pair of functional equations in a general framework using an auxiliary game: the splitting game, introduced by Sorin (A First Course on Zero-Sum Repeated Games, preprint).

关键词： incomplete information repeated games variational inequalities functional equations convexification operator convex-concave functions

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METHOD FOR MINIMIZING A convex-concave FUNCTION OVER A convex SET

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1991年第2期70卷 377-384页

作者： MUU, LD OETTLI, W INST MATH HANOIVIETNAM

A branch-and-bound method is proposed for minimizing a convex-concave function over a convex set. The minimization of a DC-function is a special case, where the subproblems connected with the bounding operation can be... 详细信息

关键词： GLOBAL OPTIMIZATION BRANCH-AND-BOUND METHODS convex-concave functions DC-PROBLEMS

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On the Anti-Monotonicity of Differential Mappings Connected with General Equilibrium problem

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Optimization 1988年第5期19卷 693-709页

作者： Uryas’Ev, S.P. W. M. Glushkov Institute of Cybernetics Kiev Ukraine

Let X be a subset of a Hilbert space H and Ψ: X x X →R, Ψ(x, x) = 0 for all x ϵ X. Let G(x) =∂yΨ(x, y) [y=x denote generalized differential with respect to the second argument at tire point (x, x). We shall be concerned with the properties of the function F sufficient to ensure the anti-monotonicity of the map G(x). It will be shown that for the anti-monotonicity of the map G(x) it is sufficient to assume convexity-concavity of the function V. In the case of the weakly convex-concave function F the map) G(x) is antimonotone under some conditions on the remainder terms. In the case of the quasieon vex- concave function F, the condition similar to the anti-monotonicity condition holds. Furthermore, some properties of weakly convex functions will be proved. © 1988, Taylor & Francis Group, LLC. All rights reserved.

关键词： Anti-mozotonie mapping convex-concave functions Equilibrium problem Variational inequalities

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On the Convergence of the Proximal Algorithm for Saddle-Point Problems

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Optimization 1995年第3期33.0卷 191-200页

作者： Moudafi, A. LACO URA 1586 Université de Limoges Limoges Cedex 87060 France

The proximal algorithm for saddle-point problems minx ∈X maxy ∈YL(x, y), where X, Y are Hilbert spaces and L: X × Y → R is a proper, closed convex-concave function in X × Y is considered, Under a minimal ... 详细信息

The proximal algorithm for saddle-point problems min_x _∈X max_y _∈YL(x, y), where X, Y are Hilbert spaces and L: X × Y → R is a proper, closed convex-concave function in X × Y is considered, Under a minimal assumption on L, δ:= inf sup L= sup inf L, it is proved that the proximmal point algorithm (PPA), with positive parameters {λ_n}, converges in general provided that (Formula Omitted). © 1995 Gordon and Breach Science Publishers SA. All rights reserved.

关键词： convex-concave functions Proximal point algorithm Saddle-points

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