Abstract
The optimality system of a quasi-variational inequality can be reformulated as a non-smooth equation or a constrained equation with a smooth function. Both reformulations can be exploited by algorithms, and their convergence to solutions usually relies on the nonsingularity of the Jacobian, or the fact that the merit function has no nonoptimal stationary points. We prove new sufficient conditions for the absence of nonoptimal constrained or unconstrained stationary points that are weaker than some known ones. All these conditions exploit some properties of a certain matrix, but do not require the nonsingularity of the Jacobian. Further, we present new necessary and sufficient conditions for the nonsingularity of the Jacobian that are based on the signs of certain determinants. Additionally, we consider generalized Nash equilibrium problems that are a special class of quasi-variational inequalities. Exploiting their structure, we also prove some new sufficient conditions for stationarity and nonsingularity results.
Highlights
IntroductionThis problem class was introduced in the paper series [1,2,3]
Since several algorithms for Generalized Nash equilibrium problems (GNEPs) have been introduced in the recent years, we discuss some of our issues for special cases of GNEPs, which are typically in a subclass of quasi-variational inequalities (QVIs) that is not easy to solve in practice
We presented a number of necessary and sufficient conditions for the main issues, when solving QVIs via a smooth constrained or a nonsmooth unconstrained equation reformulation of their Karush– Kuhn–Tucker (KKT) conditions
Summary
This problem class was introduced in the paper series [1,2,3]. It generalizes the classical variational inequality by allowing dependence of the feasible set from the point under consideration. We will discuss the central conditions to ensure that general algorithms for QVIs are well defined and converge. These are conditions guaranteeing that stationary points of a merit function are solutions, and conditions guaranteeing nonsingularity of some Jacobian matrix in order to compute the descent direction. Since several algorithms for GNEPs have been introduced in the recent years, we discuss some of our issues for special cases of GNEPs, which are typically in a subclass of QVIs that is not easy to solve in practice
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