On the use of consistent approximations in the solution of semi-infinite optimization and optimal control problems
type: Item
instance of: scholarly article
title: On the use of consistent approximations in the solution of semi-infinite optimization and optimal control problems
zbMATH Open document ID: 0805.90107
DOI: 10.1007/BF01585175
author: Elijah Polak
published in: Mathematical Programming. Series A. Series B
publication date: 1995-01-31T00:00:00Z
review text: We consider a pair consisting of an optimization problem and its optimality function \((P,\theta)\), and define consistency of approximating problem-optimality function pairs, \((P_ N,\theta_ N)\) to \((P,\theta)\), in terms of the epigraphical convergence of the \(P_ N\) to \(P\), and the hypographical convergence of the optimality functions \(\theta_ N\) to \(\theta\). We then show that standard discretization techniques decompose semi-infinite optimization and optimal control problems into families of finite-dimensional problems, which, together with associated optimality functions, are consistent discretizations to the original problems. We then present two types of techniques for using consistent approximations in obtaining an approximate solution of the original problems. The first is a ``filter'' type technique, similar to that used in conjunction with penalty functions, the second one is an adaptive discretization technique that can be viewed as an implementation of a conceptual algorithm for solving the original problems.
Mathematics Subject Classification ID: 90C34
zbMATH DE Number: 558702
zbMATH Keywords: optimal control
zbMATH Keywords: epiconvergence
zbMATH Keywords: consistency
zbMATH Keywords: epigraphical convergence
zbMATH Keywords: discretization
MaRDI profile type: Publication
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cites work: On the Optimal Control of Systems Described by Evolution Equations
cites work: A Tool for the Analysis of Quasi-Newton Methods with Application to Unconstrained Minimization
cites work: Optimization and nonsmooth analysis
cites work: Q5629139
cites work: Convergence of Functions: Equi-Semicontinuity
cites work: Diagonally Modified Conditional Gradient Methods for Input Constrained Optimal Control Problems
cites work: Function minimization by conjugate gradients
cites work: Q5584384
cites work: Effective diagonalization strategies for the solution of a class of optimal design problems
cites work: An Adaptive Precision Gradient Method for Optimal Control
cites work: A Pointwise Quasi-Newton Method for Integral Equations
cites work: Q3967968
cites work: The Fritz John necessary optimality conditions in the presence of equality and inequality constraints
cites work: Q5672476
cites work: Q3972973
cites work: Rate-Preserving Discretization Strategies for Semi-Infinite Programming and Optimal Control
cites work: On the Extension of Newton’s Method to Semi-Infinite Minimax Problems
cites work: Rates of convergence for adaptive Newton methods
description: scientific article; zbMATH DE number 558702