Nonsmooth optimization reformulations characterizing all solutions of jointly convex generalized Nash equilibrium problems

Abstract

Generalized Nash equilibrium problem, Jointly convex, Optimization reformulation, Continuity, PC 1 mapping, Semismoothness, Constant rank constraint qualification,

Generalized Nash Equilibrium Problems

“…Under suitable conditions, bothV γ andV αβ are piecewise smooth (likeV itself, they are usually nonsmooth and possibly even discontinuous). Hence their degree of smoothness is less than the one of V γ and V αβ , however, the corresponding optimization problems have the advantage that their solution characterize all equilibria of a GNEP (not just the variational equilibria), see Dreves and Kanzow (2009) for more details.…”

Gap functions for quasi-equilibria

“…Gap functions have been originally conceived for variational inequalities [15,29] and later extended to EPs [48,49], QVIs [32,19,30,59,2,33,35], jointly convex GNEPs via the NikaidoIsoda binfunction [37,38,18,56] and generic GNEPs via QVI reformulations [46]. Though descent type methods based on gap functions have been extensively developed for EPs (see, for instance, [49,44,12,43,45,6,8,9] and Section 3.2 in the survey paper [7]), the analysis of gap functions for QVIs is focused on smoothness properties [19,30,59,18,35] and error bounds [2,33] while no algorithm is developed.…”

“…Though descent type methods based on gap functions have been extensively developed for EPs (see, for instance, [49,44,12,43,45,6,8,9] and Section 3.2 in the survey paper [7]), the analysis of gap functions for QVIs is focused on smoothness properties [19,30,59,18,35] and error bounds [2,33] while no algorithm is developed. A descent method has been developed in [38] for jointly convex GNEPs; anyway, the choice of restricting to the computation of normalized equilibria makes the problem actually fall within the EP (and not the QEP) framework.…”

Nonsmooth optimization reformulations of player convex generalized Nash equilibrium problems

“…So far there is a very limited number of papers that deal with the player convex case of a GNEP, see [6][7][8]10,14,24,25]. These papers take different approaches using penalty methods, variational inequalities, and KKT systems.…”

“…More specifically, the current work may be viewed as an extension of the two previous papers [31,5]. In [31], some optimization reformulations of the jointly convex GNEP were considered, with a particular emphasis on differentiable formulations.…”