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Local behavior of an iterative framework for generalized equations with nonisolated solutions. (English) Zbl 1023.90067

Summary: An iterative framework for solving generalized equations with nonisolated solutions is presented. For generalized equations with the structure \(0\in F(z)+{\mathcal T}(z)\), where \({\mathcal T}\) is a multifunction and \(F\) is single-valued, the framework covers methods that, at each step, solve subproblems of the type \(0\in{\mathcal A}(z,s)+T(z)\). The multifunction \({\mathcal A}\) 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 \(0\in F(z)+{\mathcal T}(z)+p\). 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.

MSC:

90C31 Sensitivity, stability, parametric optimization
49J53 Set-valued and variational analysis
65K10 Numerical optimization and variational techniques
90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
90C48 Programming in abstract spaces
65H10 Numerical computation of solutions to systems of equations
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