Topological fixed point theory and applications to variational inequalities

Abstract

This is the first part of a work on generalized variational inequalities and their applications in optimization. It proposes a general theoretical framework for the solvability of variational inequalities with possibly non-convex constraints and objectives. The framework consists of a generic constrained nonlinear inequality ( $\exists\hat{u}\in\Psi(\hat {u})$ , $\exists \hat{y}\in\Phi(\hat{u})$ , with $\varphi(\hat{u},\hat{y},\hat{u})\leq \varphi(\hat{u},\hat{y},v)$ , $\forall v\in\Psi(\hat{u})$ ) derived from new topological fixed point theorems for set-valued maps in the absence of convexity. Simple homotopical and approximation methods are used to extend the Kakutani fixed point theorem to upper semicontinuous compact approachable set-valued maps defined on a large class of non-convex spaces having non-trivial Euler-Poincare characteristic and modeled on locally finite polyhedra. The constrained nonlinear inequality provides an umbrella unifying and extending a number of known results and approaches in the theory of generalized variational inequalities. Various applications to optimization problems will be presented in the second part to this work to be published ulteriorly.