Abstract.
An iterative framework for solving generalized equations with nonisolated solutions is presented. For generalized equations with the structure , where is a multifunction and F is single-valued, the framework covers methods that, at each step, solve subproblems of the type . The multifunction 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 . 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.
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Received: November 2001 / Accepted: November 2002 Published online: December 9, 2002
Key Words. generalized equation – nonisolated solutions – Newton's method – superlinear convergence – upper Lipschitz-continuity – mixed complementarity problem – error bounds
Mathematics Subject Classification (1991): 90C30, 65K05, 90C31, 90C33
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Fischer, A. Local behavior of an iterative framework for generalized equations with nonisolated solutions. Math. Program., Ser. A 94, 91–124 (2002). https://doi.org/10.1007/s10107-002-0364-4
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DOI: https://doi.org/10.1007/s10107-002-0364-4