Stable sequential Kuhn-Tucker theorem in iterative form or a regularized Uzawa algorithm in a regular nonlinear programming problem
type: Item
instance of: scholarly article
title: Stable sequential Kuhn-Tucker theorem in iterative form or a regularized Uzawa algorithm in a regular nonlinear programming problem
zbMATH Open document ID: 1327.90327
DOI: 10.1134/S0965542515060111
author: M. I. Sumin
published in: Computational Mathematics and Mathematical Physics
publication date: 2015-09-29T00:00:00Z
review text: The paper of M. I. Sumin is a valuable contribution to the analytic theory and numerical mathematics of Mathematical and Continuous Optimization and of Optimal Control Theory which have become a core technology in various disciplines, especially, in Sciences and Engineering, in Economics, Finance and OR. It pays special attention to the noisy or uncertain nature of initial data and to a constructive use of regularization techniques. In fact, the author investigates a parametric nonlinear programming problem in a metric space with an operator equality constraint in a Hilbert space presuming that its lower semicontinuous value function at a chosen individual parameter value has particular properties of subdifferentiability in the sense of nonlinear and nonsmooth analysis. That subdifferentiability can be interpreted as the existence of a proximal subgradient or a Fréchet subdifferential. This means that an individual problem has a corresponding generalized Kuhn-Tucker vector. Given that assumption, a stable subsequential Kuhn-Tucker Theorem in nonsmooth iterative form is demonstrated and discussed in terms on minimizing sequences, essentially, through the dual regularization method. This result provides necessary and sufficient conditions for a stable construction of a minimizing approximate solution in Warga's sense for the considered problem, whose initial data can be specified approximately. A core difference of the theorem proved from its classical analogue of the same name consists in the fact that the former one takes into account the possible sensitivity of the problem under perturbations of the initial data and, therefore, permits the inherited instability of classical optimality conditions. That result can be treated as a regularized extension of the well-known Uzawa algorithm to nonlinear programming problems. Eventually, the theorem is applied to the ``easiest'' nonlinear optimal control problem, which is a time optimal control problem. This excellent artic…
Mathematics Subject Classification ID: 90C31
Mathematics Subject Classification ID: 90C48
Mathematics Subject Classification ID: 90C46
zbMATH DE Number: 6486283
zbMATH Keywords: nonlinear programming
zbMATH Keywords: parametric problem
zbMATH Keywords: perturbation method
zbMATH Keywords: stability
zbMATH Keywords: sequential optimization
zbMATH Keywords: minimizing sequence
zbMATH Keywords: duality
zbMATH Keywords: regularization
zbMATH Keywords: Lagrange principle Kuhn-Tucker Theorem in nondifferential form
zbMATH Keywords: Pontyagin's maximum principle
reviewed by: Gerhard-Wilhelm Weber
MaRDI profile type: Publication
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full work available at URL: s0965542515060111
OpenAlex ID: W916687805
description: scientific article; zbMATH DE number 6486283