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Optimization methods and stability of inclusions in Banach spaces. (English) Zbl 1158.49007

Let \(X\) and \(P\) be Banach spaces, \(p\in P\) and \(F:X\rightrightarrows P\) be a multifunction with closed graph. This paper deals with the generalized equation consisting in finding \(x\) such that \(p\in F(x)\). The main contribution is to show the close relationship existing between local Lipschitz stability and convergence of several algorithmic schemes, which encompass descent methods, approximate projections, penalizations, successive approximations and Newton methods. Nonlinear perturbed inclusions of the type \(p\in h(x)+F(x),\) with \(h:X\to P\) being a nonlinear perturbation, are also considered.

MSC:

49J27 Existence theories for problems in abstract spaces
49K40 Sensitivity, stability, well-posedness
90C31 Sensitivity, stability, parametric optimization
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