A parametric version of the Borwein-Preiss smooth variational principle is presented, which states that under suitable assumptions on a given convex function depending on a parameter, the minimum point of a smooth convex perturbation of it depends continuously on the parameter. Some applications are given: existence of a Nash equilibrium and a solution of a variational inequality for a system o...

This paper provides a complete solution to the problem of minimax inequality for an arbitrary topological space that may be discrete, continuum, non-compact or non-convex. We establish a single condition, γ-recursive transfer lower semicontinuity, which is necessary and sufficient for the existence of equilibrium of minimax inequality without imposing any kind of convexity nor any restriction o...

We consider the problem of optimality, in a minimax sense, and adaptivity to the margin and to regularity in binary classification. We prove an oracle inequality, under the margin assumption (low noise condition), satisfied by an aggregation procedure which uses exponential weights. This oracle inequality has an optimal residual: (logM/n) where κ is the margin parameter, M the number of classif...

The problem ofgeneralized equilibrium problem is very general in the different subjects .Optimization problems, variational inequalities, Nash equilibrium problem and minimax problems are as special cases of generalized equilibrium problem. The purpose of this paper is to investigate the problem of approximating a common element of the set of generalized equilibrium problem, variational inequal...

The theory of the gap functions is extended to the variational inequality introduced by Minty. Exploiting the minimax formulation of a variational inequality, a new class of gap functions is defined. Descent methods,based on the minimization of the new class of gap functions, are analysed.

In this technical note, we give two extensions of the classical Fano inequality in information theory. The first extends Fano’s inequality to the setting of estimation, providing lower bounds on the probability that an estimator of a discrete quantity is within some distance t of the quantity. The second inequality extends our bound to a continuum setting and provides a volume-based bound. We i...

Robust synthesis is one of the remaining challenges in model predictive control (MPC). One way to robustify an MPC controller is to formulate a minimax problem, i.e., optimize a worst-case performance measure. For systems modeled with an uncertain gain, there are many results available. Typically, the minimax formulations have given intractable problems, or unorthodox performance measures have ...