On error bounds and Newton-type methods for generalized Nash equilibrium problems

Abstract

Error bounds (estimates for the distance to the solution set of a given problem) are key to analyzing convergence rates of computational methods for solving the problem in question, or sometimes even to justifying convergence itself. That said, for the generalized Nash equilibrium problems (GNEP), the theory of error bounds had not been developed in depth comparable to the fields of optimization and variational problems. In this paper, we provide a systematic approach which should be useful for verifying error bounds for both specific instances of GNEPs and for classes of GNEPs. These error bounds for GNEPs are based on more general results for constraints that involve complementarity relations and cover those (few) GNEP error bounds that existed previously, and go beyond. In addition, they readily imply a Lipschitzian stability result for solutions of GNEPs, a subject where again very little had been known. As a specific application of error bounds, we discuss Newtonian methods for solving GNEPs. While we do not propose any significantly new methods in this respect, some new insights into applicability to GNEPs of various approaches and into their convergence properties are presented.

Convergence conditions for Newton-type methods applied to complementarity systems with nonisolated solutions

“…Here, we give a different argument which is, in a sense, more direct: we demonstrate that Condition 5 implying the whole set of Assumptions 1-4 holds under noncriticality of the Lagrange multiplier. Observe that the error bound itself is an immediate consequence of Proposition 6.1 below and of [20,Lemma 1].…”

“…Condition 5 below was already considered in [20] and is equivalent to Condition 4 for problem (1.3) with Ω given by (1.4).…”

“…As is explained in [20], if, for each partition (I 1 , I 2 ) of I 0 , some constraint qualification implying the local error bound condition for (5.2) is satisfied at z * , then Condition 5 holds. For example, if MFCQ or RCRCQ holds at z * for system (5.2) for each partition (I 1 , I 2 ) of I 0 , then Condition 5 is valid.…”

“…For example, if MFCQ or RCRCQ holds at z * for system (5.2) for each partition (I 1 , I 2 ) of I 0 , then Condition 5 is valid. In [20], it is actually stated that Condition 5 implies Condition 1 for system (1.3) and so Assumption 2. However, the relation between Condition 5 and Assumption 3 was not analyzed in [20].…”

“…Sections 7 and 8 consider different reformulations of KKT systems associated to generalized Nash equilibrium problems, where we exhibit sufficient conditions for the required error bounds in that setting. Specifically, a result from [20] based on the full row rank of a certain matrix is recovered. In addition, a more general result is presented, showing that even the weaker constant rank of certain matrices is sufficient.…”