### Giancarlo Bigi (University of Pisa, Italy)

#### Descent methods for equilibria

*Wednesday 18 September 2013 at 15.30, JCMB 6206*

##### Abstract

The so-called equilibrium problem is a mathematical model, which subsumes,
among others, variational inequalities, noncooperative games, multi-objective
and inverse optimization in a unified framework.

Descent methods exploit optimization reformulations of equilibrium problems
(EPs) through appropriate gap functions. The talk aims at providing an
overview of this class of methods. The basic ideas and features will be
illustrated, in particular the role of monotonicity in overcoming the
nonconvexities which come into play will be underlined. Afterwards, I will
focus on some recent developments. First, I will show an algorithm for EPs
with nonlinear constraints which involves only linearly constrained
optimization problems: polyhedral approximations of the feasible region are
exploited together with possibly unfeasible search directions, which call for
exact penalties. Finally, I will show an algorithm which exploits pairs of gap
functions in such a way that search directions can be exploited with no
feasibility concern.

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