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Title: A Cutting Surface Algorithm for Semi-Infinite Convex Programming with an Application to Moment Robust Optimization

Abstract

In this paper, we present and analyze a central cutting surface algorithm for general semi-infinite convex optimization problems and use it to develop a novel algorithm for distributionally robust optimization problems in which the uncertainty set consists of probability distributions with given bounds on their moments. Moments of arbitrary order, as well as nonpolynomial moments, can be included in the formulation. We show that this gives rise to a hierarchy of optimization problems with decreasing levels of risk-aversion, with classic robust optimization at one end of the spectrum and stochastic programming at the other. Although our primary motivation is to solve distributionally robust optimization problems with moment uncertainty, the cutting surface method for general semi-infinite convex programs is also of independent interest. The proposed method is applicable to problems with nondifferentiable semi-infinite constraints indexed by an infinite dimensional index set. Examples comparing the cutting surface algorithm to the central cutting plane algorithm of Kortanek and No demonstrate the potential of our algorithm even in the solution of traditional semi-infinite convex programming problems, whose constraints are differentiable, and are indexed by an index set of low dimension. After the rate of convergence analysis of the cutting surface algorithm, we extend themore » authors' moment matching scenario generation algorithm to a probabilistic algorithm that finds optimal probability distributions subject to moment constraints. The combination of this distribution optimization method and the central cutting surface algorithm yields a solution to a family of distributionally robust optimization problems that are considerably more general than the ones proposed to date.« less

Authors:
 [1];  [1]
  1. Northwestern Univ., Evanston, IL (United States)
Publication Date:
Research Org.:
Northwestern Univ., Evanston, IL (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); National Science Foundation (NSF)
OSTI Identifier:
1321089
Grant/Contract Number:  
SC0005102; CMMI-1100868; SP0011568
Resource Type:
Accepted Manuscript
Journal Name:
SIAM Journal on Optimization
Additional Journal Information:
Journal Volume: 24; Journal Issue: 4; Journal ID: ISSN 1052-6234
Publisher:
SIAM
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; semi-infinite programming; robust optimization; distributionally robust optimization; stochastic programming; moment matching; column generation; cutting surface methods; cutting plane methods; moment problem

Citation Formats

Mehrotra, Sanjay, and Papp, Dávid. A Cutting Surface Algorithm for Semi-Infinite Convex Programming with an Application to Moment Robust Optimization. United States: N. p., 2014. Web. https://doi.org/10.1137/130925013.
Mehrotra, Sanjay, & Papp, Dávid. A Cutting Surface Algorithm for Semi-Infinite Convex Programming with an Application to Moment Robust Optimization. United States. https://doi.org/10.1137/130925013
Mehrotra, Sanjay, and Papp, Dávid. Thu . "A Cutting Surface Algorithm for Semi-Infinite Convex Programming with an Application to Moment Robust Optimization". United States. https://doi.org/10.1137/130925013. https://www.osti.gov/servlets/purl/1321089.
@article{osti_1321089,
title = {A Cutting Surface Algorithm for Semi-Infinite Convex Programming with an Application to Moment Robust Optimization},
author = {Mehrotra, Sanjay and Papp, Dávid},
abstractNote = {In this paper, we present and analyze a central cutting surface algorithm for general semi-infinite convex optimization problems and use it to develop a novel algorithm for distributionally robust optimization problems in which the uncertainty set consists of probability distributions with given bounds on their moments. Moments of arbitrary order, as well as nonpolynomial moments, can be included in the formulation. We show that this gives rise to a hierarchy of optimization problems with decreasing levels of risk-aversion, with classic robust optimization at one end of the spectrum and stochastic programming at the other. Although our primary motivation is to solve distributionally robust optimization problems with moment uncertainty, the cutting surface method for general semi-infinite convex programs is also of independent interest. The proposed method is applicable to problems with nondifferentiable semi-infinite constraints indexed by an infinite dimensional index set. Examples comparing the cutting surface algorithm to the central cutting plane algorithm of Kortanek and No demonstrate the potential of our algorithm even in the solution of traditional semi-infinite convex programming problems, whose constraints are differentiable, and are indexed by an index set of low dimension. After the rate of convergence analysis of the cutting surface algorithm, we extend the authors' moment matching scenario generation algorithm to a probabilistic algorithm that finds optimal probability distributions subject to moment constraints. The combination of this distribution optimization method and the central cutting surface algorithm yields a solution to a family of distributionally robust optimization problems that are considerably more general than the ones proposed to date.},
doi = {10.1137/130925013},
journal = {SIAM Journal on Optimization},
number = 4,
volume = 24,
place = {United States},
year = {2014},
month = {10}
}

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