Perturbation theory for mathematical programming problems
type: Item
instance of: scholarly article
title: Perturbation theory for mathematical programming problems
zbMATH Open document ID: 0568.90089
DOI: 10.1007/BF00941069
published in: Journal of Optimization Theory and Applications
publication date: 1986-01-01T00:00:00Z
review text: Mathematical programming (MP) problems depending on a small parameter are investigated. Attention is paid to the cases where the solutions to the reduced program and/or the solutions to the dual reduced program are not unique. Conditions are given for the convergence of perturbed solutions to a point of the reduced problem solution set, if the small parameter tends to zero. It is shown how to find this point and how to construct an approximate solution to the perturbed program. A singular situation may appear if the dual solution set is unbounded. In this case, a gap between perturbed and reduced solutions may arise. However, it is shown that the perturbed solutions are close to the solutions of some modified reduced problem. The fractional usefulness of perturbation theory is demonstrated by considering two LP problems. Decomposition and aggregation procedures are constructed on the base of general results to find suboptimal solutions of these problems.
Mathematics Subject Classification ID: 90C31
Mathematics Subject Classification ID: 65K05
Mathematics Subject Classification ID: 90C05
zbMATH DE Number: 3908188
zbMATH Keywords: perturbation
zbMATH Keywords: singularities
zbMATH Keywords: approximate solution
zbMATH Keywords: Decomposition
zbMATH Keywords: aggregation
zbMATH Keywords: suboptimal solutions
author: V. Gaitsgory
author: Anatoli A. Pervozvanski
MaRDI profile type: Publication
cites work: Q3236246
cites work: Differential stability in infinite-dimensional nonlinear programming
cites work: Differential properties of the marginal function in mathematical programming
cites work: Mathematical programming. (Matematicheskoe programmirovanie)
cites work: Q3945945
cites work: Q5588268
cites work: Q4109500
cites work: Objective function approximations in mathematical programming
cites work: Nonlinear Perturbation of Linear Programs
cites work: The Fritz John necessary optimality conditions in the presence of equality and inequality constraints
cites work: Convex Analysis
full work available at URL: https://doi.org/10.1007/bf00941069
OpenAlex ID: W1982541069
description: scientific article; zbMATH DE number 3908188