Recently Auslender & Cronzeix introduced a well-behaved asymptotical notion for convexfunctions. The aim of this work is to extend this notion to the saddle case.
Recently Auslender & Cronzeix introduced a well-behaved asymptotical notion for convexfunctions. The aim of this work is to extend this notion to the saddle case.
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-poi...
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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.
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 traject...
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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/concavefunctions and constraints, and - as a special case - to projected saddle-point dynamics. (C) 2017 Elsevier B.V. All rights reserved.
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 ...
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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.
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 per...
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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).
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...
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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 solved effectively.
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 con...
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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 ...
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