Coderivatives in parametric optimization
The paper is devoted to sensitivity analysis of constrained optimization problems and the main attention is paid to the study of Lipschitzian stability of solution maps in optimization problems (particularly stationary point multifunctions and stationary point-multiplier multifunctions in parametric mathematical programming). The basic tools involve derivative-like objects for set-valued mappings, called coderivatives, which give extensions of the classical adjoint derivative operator and play a role in characterizations of Lipschitzian behaviour and metric regularity for general multifunctions. The authors focus on a family of parametrized minimization problems where the sum of the parametrized objective function plus a parametrized function that incorporates the constraints (it is called the constraint function) is minimized. The constraint function is lower semicontinuous and extended real-valued. The authors use coderivatives to analyze stationary point multifunctions, which represent the different stationary points associated with each parameter value, by developing a general implicit mapping theorem in order to get an estimate for the coderivative of the stationary point multifunction in terms of the coderivative of the partial subgradient mapping corresponding to the constraint function. A general partial coderivative estimate in terms of the full coderivative is developed, which results in an estimate in terms of the second-order subdifferential of the constraint function. The paper focus attention to the estimation of such second-order sub\-differential, paying particular attention on the case when the constraint function is a special kind of composite function called ``strongly amenable'', which means a composite of a convex, lower semicontinuous function with a two times continuously differentiable mapping on a neighbourhood of the point \(x\) and the parameter \(w\) that fulfilled a constraint qualification which for nonlinear problems coincides with the Mangasarian-Fromovitz constraint qualification. These cover many interesting examples including standard nonlinear programs, and the final estimates are given in terms of standard derivative conditions on the original data of the problem. In the penultimate section the authors study stationary point-multiplier pairs and make connections to previous results on stationary points. Finally, the paper studies the special case when canonical perturbations are present in the parametrization of the optimization model.
scientific article; zbMATH DE number 2077961
scientific article