Gap Function Approach to the Generalized Nash Equilibrium Problem

Abstract

Abstract. We consider an optimization reformulation approach for the generalized Nash equilibrium problem (GNEP) that uses the regularized gap function of a quasi-variational inequality (QVI). The regularized gap function for QVI is in general not differentiable, but only directionally differentiable. Moreover, a simple condition has yet to be established, under which any stationary point of the regularized gap function solves the QVI. We tackle these issues for the GNEP in which the shared constraints are given by linear equalities, while the individual constraints are given by convex inequalities. First, we formulate the minimization problem involving the regularized gap function, and show the equivalence to GNEP. Next, we establish the differentiability of the regularized gap function and show that any stationary point of the minimization problem solves the original GNEP under some suitable assumptions. Then, by using a barrier technique, we propose an algorithm that sequentially solves minimization problems obtained from GNEPs with the shared equality constraints only. Further, we discuss the case of shared inequality constraints and present an algorithm that utilizes the transformation of the inequality constraints to equality constraints by means of slack variables. We present some results of numerical experiments to illustrate the proposed approach.

Noncooperative Day-Ahead Bidding Strategies for Demand-Side Expected Cost Minimization With Real-Time Adjustments: A GNEP Approach

“…They can be formulated as quasi-variational inequality (QVI) problems [19]; however, in spite of some interesting and promising recent advancements (see, e.g., [26], [27]), no efficient numerical methods based on the QVI reformulation have been developed yet. Nevertheless, for this type of GNEPs, some VI techniques can still be employed [17].…”

“…A heuristic procedure to deal with such cases is presented in Appendix II-B. These considerations also apply to condition (27) given in Theorem 1(a.2).…”

Gap functions for quasi-equilibria

“…Furthermore, the same equivalence has been exploited in [46] in a very particular framework: the QVI reformulation of a generalized Nash equilibrium problem with only linear equality shared constraints. Indeed, linear equality shared constraints satisfy both the active ∇-monotonicity condition (9) and the additional assumption on y(x) (see the last paragraph of this section).…”

“…Also Newton type methods, which guarantee only local convergence, have been developed [53,54]. A way to study GNEPs is to formulate them as QVIs (see, for istance, [34]) and exploit the corresponding theories and algorithms: projection methods are expolited in [60] while penalty techniques and barrier methods in [46]. The simultaneous resolution of the KKT conditions of the optimization problems describing a GNEP has been carried out through locally convergent Newton type techniques [21,42] and globally convergent interior-point type techniques [17].…”

“…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.…”