Local behavior of an iterative framework for generalized equations with nonisolated solutions

An iterative framework for solving generalized equations with nonisolated solutions is presented. For generalized equations with the structure 0 ∈ F(z) + T (z), where T is a multifunction and F is singlevalued, the framework covers methods that, at each step, solve subproblems of the type 0 ∈ A(z, s)+ T (z). The multifunction A approximates F around s. Besides a condition on the quality of this approximation, two other basic assumptions are employed to show Q-superlinear or Q-quadratic convergence of the iterates to a solution. A key assumption is the upper Lipschitz-continuity of the solution set map of the perturbed generalized equation 0 ∈ F(z) + T (z) + p. Moreover, the solvability of the subproblems is required. Conditions that ensure these assumptions are discussed in general and by means of several applications. They include monotone mixed complementarity problems, Karush-Kuhn-Tucker systems arising from nonlinear programs, and nonlinear equations. Particular results deal with error bounds and upper Lipschitz-continuity properties for these problems.