Optimality Conditions and Constraint Qualifications for Generalized Nash Equilibrium Problems and Their Practical Implications

Abstract

Generalized Nash Equilibrium Problems (GNEPs) are a generalization of the classic Nash Equilibrium Problems (NEPs), where each player's strategy set depends on the choices of the other players. In this work we study constraint qualifications and optimality conditions tailored for GNEPs and we discuss their relations and implications for global convergence of algorithms. Surprisingly, differently from the case of nonlinear programming, we show that, in general, the KKT residual can not be made arbitrarily small near a solution of a GNEP. We then discuss some important practical consequences of this fact. We also prove that this phenomenon is not present in an important class of GNEPs, including NEPs. Finally, under a weak constraint qualification introduced, we prove global convergence to a KKT point of an Augmented Lagrangian algorithm for GNEPs and under the quasinormality constraint qualification for GNEPs, we prove boundedness of the dual sequence.

A Sequential Optimality Condition Related to the Quasi-normality Constraint Qualification and Its Algorithmic Consequences

“…Actually, we have also not established this result in the first publicly available version of this paper, although it is a natural consequence of our theory. Very recently, and citing this earlier version, Bueno, Haeser and Rojas reported this fact in a more general context, namely, of the generalized Nash equilibrium problems (see [16,Theorem 6.3]). In this sense, it is not the first time that Corollary 4.3 appears in the literature.…”

“…according to [15]. In this paper, we only deal with the augmented Lagrangian (1) (and then we always have (16)), but we note that the general form (15) is also appropriate for the case when a non-quadratic penalty augmented Lagrangian function is employed, as in [18,29]. It is known that when Algorithm 1 does not stop by failure, it generates an AKKT sequence for the problem (P) if its limit point is feasible (see [15]).…”

An Augmented Lagrangian Method for Cardinality-Constrained Optimization Problems

“…For NLPs it has been shown that this variant possesses strong convergence properties even under very mild assumptions [4,6]. It has since been applied to solve various other problems, including MPCC [7,26], quasi-variational inequalities [27], generalized Nash equilibrium problems [16,29], and semidefinite programming [8,14].…”

Analytical solution of quasi-equilibrium problems in one variable

“…La literatura sobre estos dos tipo de problemas asi como para EPs es amplia tanto desde el punto de vista de existencia y unicidad de soluciones, de algoritmos para hallar soluciones y de modelaje de problemas que surgen en diferentesáreas, para QVIs ver por ejemplo [9,27,18] y referencias, para GNEPs [17,16,15,14,7] y referencias, para EPs [30,33,32,6,31] y referencias. Entanto el estudio de QEP ha recibido menos atención en comparación con los problemas antes mencionados, esto puede ser debido a la complejidad y generalidad de su formulación y de aun no encontrarse suficientes problemas aplicativos que puedan ser modelados estrictamente por QEPs, y que no puedan ser modelados por algun problema que sea un caso particular de un QEP.…”