Gap functions for quasi-equilibria

Abstract

An approach for solving quasi-equilibrium problems (QEPs) is proposed relying on gap functions, which allow reformulating QEPs as global optimization problems. The (generalized) smoothness properties of a gap function are analysed and an upper estimates of its Clarke directional derivative is given. Monotonicity assumptions on both the equilibrium and constraining bifunctions are a key tool to guarantee that all the stationary points of a gap function actually solve QEP. A few classes of constraints satisfying such assumptions are identified covering a wide range of situations. Relying on these results, a descent method for solving QEP is devised and its convergence proved. Finally, error bounds are given in order to guarantee the boundedness of the sequence generated by the algorithm.

Error bounds for generalized mixed weak vector quasiequilibrium problems via regularized gap functions

“…Motivated by Fukushima (1992), based on strong monotonicity assumptions Yamashita & Fukushima (1997) studied global error bounds for general variational inequalities under using regularized gap functions of the Moreau-Yosida type. Since then, the study of error bounds for related-optimization problems has become an interesting and important topic in optimization theory (see Husain & Singh (2017), Khan & Chen (2015), Yamashita & Fukushima (1997), Bigi & Passacantando (2016), Fukushima (1992), Anh, Hung, & Tam (2018), Mastroeni (2003) and the references therein). In Khan & Chen (2015), the regularized gap functions of Fukushima type versions and error bounds were studied for generalized mixed vector equilibrium problems infinite-dimensional spaces.…”

Algorithms for generalized potential games with mixed-integer variables

“…We consider the jointly convex linear GNEP with mixed-integer variables defined by problems (6). This generalized potential game is particularly relevant if the number of the firms is small, and challenging if the number of decision variables of each firm is large.…”

Error bound analysis for vector equilibrium problems with partial order provided by a polyhedral cone

“…Using gap functions in forms of the Fukushima regularization and the Moreau-Yosida regularization, Yamashita and Fukushima [39] established global error bounds for general variational inequalities. Thereafter, many regularized gap functions and error bounds have been studied for various kinds of equilibrium problems and variational inequalities, see e.g., [1,3,19,[21][22][23][24]27,35] and the references therein for a more detailed discussion of interesting topics. In particular, Khan and Chen [27] established regularized gap functions and error bounds for generalized mixed vector equilibrium problems under the partial order introduced by the usual positive cone in finite dimensional spaces.…”