Approximations and solution estimates in optimization
type: Item
instance of: scholarly article
title: Approximations and solution estimates in optimization
zbMATH Open document ID: 1410.90241
author: Johannes O. Royset
published in: Mathematical Programming. Series A. Series B
publication date: 2018-08-22T00:00:00Z
review text: The aim of this paper is to study approximations of lower semicontinuous (lsc) functions and of the solutions of optimization problems in metric spaces. The main tool is the Attouch-Wets distance of two lsc functions, which is defined in terms of their epigraphs. Among the obtained results are an estimate of the difference of the optimal values of two optimization problems, an estimate of the distance of their respective sets of solutions, an approximation of near-optimal, near-feasible solutions, an approximation of the level sets, etc. The last section is devoted to epi-splines, which are a particular kind of approximating functions defined by a finite number of parameters. These are extended real-valued, and constant on each element of a given partition of a metric space. It is shown that every lsc function can be approximated by a sequence of epi-splines, epi-converging and pointwise converging to the function. Actually, the set of rational-valued epi-splines is dense in the space of lsc functions. An estimation of the rate of convergence of epi-splines to lsc functions is obtained.
reviewed by: Nicolas Hadjisavvas
Mathematics Subject Classification ID: 90C48
Mathematics Subject Classification ID: 90C31
Mathematics Subject Classification ID: 41A65
zbMATH DE Number: 6923437
zbMATH Keywords: epi-convergence
zbMATH Keywords: Attouch-Wets distance
zbMATH Keywords: solution stability
zbMATH Keywords: near-optimality
MaRDI profile type: Publication
full work available at URL: s10107-017-1165-0
OpenAlex ID: W2401014899
cites work: Q5186236
cites work: A Convergence Theory for Saddle Functions
cites work: Isometries for the Legendre-Fenchel Transform
cites work: Quantitative Stability of Variational Systems: I. The Epigraphical Distance
cites work: Quantitative Stability of Variational Systems II. A Framework for Nonlinear Conditioning
cites work: Quantitative stability of variational systems. III: \(\varepsilon\)- approximate solutions
cites work: The topology of the \(\rho\)-Hausdorff distance
cites work: Set-valued analysis
cites work: A survey on error bounds for lower semicontinuous functions
cites work: Q3825271
cites work: Characterizations of error bounds for lower semicontinuous functions on metric spaces
cites work: Variational pairs and applications to stability in nonsmooth analysis.
cites work: Q4277838
cites work: Convex Optimization and the Epi-Distance Topology
cites work: The EPI-Distance Topology: Continuity and Stability Results with Applications to Convex Optimization Problems
cites work: A Characterization of Epi-Convergence in Terms of Convergence of Level Sets
cites work: Q4503250
cites work: 10.1162/153244302760200704
cites work: Tilt Stability, Uniform Quadratic Growth, and Strong Metric Regularity of the Subdifferential
cites work: A study of tilt-stable optimality and sufficient conditions
cites work: Error estimation in nonlinear optimization
cites work: Remarks to an equivalent formulation of ekeland’s variational principle
cites work: On metric and calmness qualification conditions in subdifferential calculus
cites work: Q4903676
cites work: Partial Smoothness, Tilt Stability, and Generalized Hessians
cites work: Characterizations of full stability in constrained optimization
cites work: Convergence of convex sets and of solutions of variational inequalities
cites work: Error bounds for lower semicontinuous functions in normed spaces
cites work: Error bounds in mathematical programming
cites work: Q3060974
cites work: Optimization. Algorithms and consistent approximations
cites work: Variational Analysis
cites work: Stability of $\varepsilon$-approximate Solutions to Convex Stochastic Programs
cites work: Fusion of hard and soft information in nonparametric density estimation
cites work: Multivariate epi-splines and evolving function identification problems
cites work: Variational Theory for Optimization under Stochastic Ambiguity
cites work: Q3976098
cites work: Q3939603
cites work: Convergence of sequences of convex sets, cones and functions
cites work: Convergence of Sequences of Convex Sets, Cones and Functions. II
cites work: On error bounds for lower semicontinuous functions.
cites work: Necessary Optimality Conditions for Optimization Problems with Variational Inequality Constraints
DOI: 10.1007/S10107-017-1165-0
description: scientific article; zbMATH DE number 6923437