13h30 - 13h55
A sequential equilibrium programming algorithm for computing quasi-equilibria
An algorithm for solving quasi-equilibrium problems (QEPs) is proposed relying on the sequential inexact resolution of equilibrium problems. First, we reformulate QEP as the fixed point problem of a set-valued map and analyse its Lipschitz continuity under strong monotonicity assumptions. Then, a few classes of QEPs satisfying these assumptions are identified. Finally, we devise an algorithm that computes an inexact solution of an equilibrium problem at each iteration and we prove its global convergence.
13h55 - 14h20
A Newton-type method for Quasi-Equilibrium problems with applications to Quasi-Variational Inequalities
We present a local fast convergence method for solving Quasi-Equilibrium Problems, QEP. We apply our results to the Generalized Nash Equilibrium Problems, GNEP that corresponds to a QEP. Also, it can be reformulated as a Quasi-Variational Problem. In the case of jointly convex GNEP, our algorithm allows finding any solutions of the problem, not only the normalized equilibrium solutions. We report numerical results showing the performance of the algorithm.
14h20 - 14h45
A Superlinearly Convergent Smoothing Newton Continuation Algorithm for Variational Inequalities over Definable Sets
We use the concept of barrier-based smoothing approximations introduced by Chua and Li [C. B. Chua and Z. Li, A barrier-based smoothing proximal point algorithm for NCPs over closed convex cones, SIAM J. Optim., 23 (2013), pp. 745–769] to extend the smoothing Newton continuation algorithm of Hayashi at el [S. Hayashi, N. Yamashita, and M. Fukushima, A combined smoothing and regularization method for monotone second-order cone complementarity problems, SIAM J. Optim., 15 (2005), pp. 593–615] to variational inequalities over general closed convex sets. We prove that when the underlying barrier has a gradient map that is definable in some o-minimal structure, the iterates generated converge superlinearly to a solution of the variational inequality. We further prove that if the convex set is proper and definable in the o-minimal expansion of the globally subanalytic sets by all power functions with real algebraic exponents, then the gradient map of its universal barrier is definable in the o-minimal expansion of the globally subanalytic sets by the exponential function. Finally, we consider the application of the algorithm to complementarity problems over epigraphs of matrix operator norm and nuclear norm, and present preliminary numerical results.
14h45 - 15h10
A stochastic weighted variational inequality in non-pivot Hilbert spaces for a transportation model with uncertainty
The talks deals with a class of stochastic weighted variational inequalities in non-pivot Hilbert spaces. In particular we present an existence result under general assumptions and we show a continuity result for the solution to strongly pseudo-monotone weighted quasi-variational inequalities. Thanks to these theoretical results, we are able to propose a new weighted transportation model with uncertainty. Furthermore, we establish the equivalence between the random weighted equilibrium principle and a stochastic weighted variational inequality. Finally, a numerical example is provided.
 A. Barbagallo, G. Scilla, Stochastic weighted variational inequalities in non-pivot Hilbert spaces with applications to a transportation model, submitted.