Optimization and Equilibrium

Partially non-convex minimax theorem and applications to remotal sets.

Event Date: Oct 18, 2023 in Optimization and Equilibrium, Seminars

Abstract: Given a convex subset B  A of a locally convex space Y  X; and a function f : Y  X ! R such that B is compact and f (y; ); y 2 Y; are concave and upper semicontinuous, we establish in a Örst step a minimax inequality of the form maxy2B infx2A f (y; x)  infx2A supy2B0 f (y; x); where B0 is the set of points y 2 B such that f (y; ) is proper and convex. The main di§erence with the classical minimax theorem is that, here, the set B0 does not need to be convex or compact. We use this result to give a new proof of the characterization of remotal sets, relying on the convexiy of the set...

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Globalization of the SCD semismooth* Newton method in nonsmooth nonconvex optimization

Event Date: Jul 19, 2023 in Optimization and Equilibrium, Seminars

Abstract:  We consider the problem of minimizing an lsc proper function. A locally superlinearly convergent method is given by the SCD (Subspace Containing Derivative) semismooth* Newton method for solving the first order necessary optimality conditions. We will discuss how to compute the SC derivative of the subdifferential mapping defining the linear system for the Newton direction. In order to globalize the SCD semismooth* Newton method, we combine it with some variant of the proximal gradient algorithm. We are able to show that every accumulation point of the sequence produced by our...

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Aproximación de juegos de campo medio de primer orden.

Event Date: May 03, 2023 in Optimization and Equilibrium, Seminario de Probabilidades de Chile, Seminars

Resumen:   Esta charla concierne la aproximación de juegos de campo medio de primer orden, o deterministas, introducidos por J.-M. Lasry y P.-L. Lions en el año 2007. Luego de introducir este tipo de juegos, nos concentraremos en la aproximación de la función valor de un jugador típico, elemento clave de la discretización del juego de campo medio. Esta última puede interpretarse como un juego de campo medio en tiempo discreto y espacio de estados finitos introducido por Gomes, Mohr y Souza en el año 2010. Luego de enunciar el teorema de convergencia principal, terminaremos la charla con...

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Convergence Rate of Nonconvex Douglas-Rachford splitting via merit functions, with applications to weakly convex constrained optimization.

Event Date: Apr 12, 2023 in Optimization and Equilibrium, Seminars

Abstract:  We analyze Douglas-Rachford splitting techniques applied to solving weakly convex optimization problems. Under mild regularity assumptions, and by the token of a suitable merit function, we show convergence to critical points and local linear rates of convergence. The merit function, comparable to the Moreau envelope in Variational Analysis, generates a descent sequence, a feature that allows us to extend to the non-convex setting arguments employed in convex optimization. A by-product of our approach is an ADMM-like method for constrained problems with weakly convex objective...

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Problem Decomposition in Convex Optimization: Advances Beyond ADMM.

Event Date: Apr 05, 2023 in Optimization and Equilibrium, Seminars

Abstract: Applications of convex optimization in areas like image processing and machine learning have stimulated a huge interest in solution methodology that can take advantage of underlying decomposable structure in a problem, especially when iterations can make good use of “prox” mappings on the problem’s components.  Very popular in this development has been the Alternating Direction Method of Multipliers (ADMM).   But other approaches that branch out from the same mathematical roots in different modes now offer new advances in the flexibility of problem formulation and...

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Percolation games.

Event Date: Mar 22, 2023 in ACGO, Optimization and Equilibrium, Seminars

Abstract: Inspired by first-passage percolation models, we consider zero-sum games on Z^d and study their limit behavior when the game duration tends to infinity. After reviewing several fundamental results in this literature, we present a generalization and discuss connections with long-term behavior of Hamilton-Jacobi equations.

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