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On smooth reformulations and direct non-smooth computations for minimax problems

Minimax problems can be approached by reformulating them into smooth problems with constraints or by dealing with the non-smooth objective directly. We focus on verified enclosures of all globally optimal points of such problems. In smooth problems in branch and bound algorithms, interval Newton met... Full description

Journal Title: Journal of Global Optimization 2013, Vol.57(4), pp.1091-1111
Main Author: Kearfott, Ralph
Other Authors: Muniswamy, Sowmya , Wang, Yi , Li, Xinyu , Wang, Qian
Format: Electronic Article Electronic Article
Language: English
Subjects:
ID: ISSN: 0925-5001 ; E-ISSN: 1573-2916 ; DOI: 10.1007/s10898-012-0014-1
Link: http://dx.doi.org/10.1007/s10898-012-0014-1
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recordid: springer_jour10.1007/s10898-012-0014-1
title: On smooth reformulations and direct non-smooth computations for minimax problems
format: Article
creator:
  • Kearfott, Ralph
  • Muniswamy, Sowmya
  • Wang, Yi
  • Li, Xinyu
  • Wang, Qian
subjects:
  • Minimax
  • Verified computations
  • Fritz John equations
ispartof: Journal of Global Optimization, 2013, Vol.57(4), pp.1091-1111
description: Minimax problems can be approached by reformulating them into smooth problems with constraints or by dealing with the non-smooth objective directly. We focus on verified enclosures of all globally optimal points of such problems. In smooth problems in branch and bound algorithms, interval Newton methods can be used to verify existence and uniqueness of solutions, to be used in eliminating regions containing such solutions, and point Newton methods can be used to obtain approximate solutions for good upper bounds on the global optimum. We analyze smooth reformulation approaches, show weaknesses in them, and compare reformulation to solving the non-smooth problem directly. In addition to analysis and illustrative problems, we exhibit the results of numerical computations on various test problems.
language: eng
source:
identifier: ISSN: 0925-5001 ; E-ISSN: 1573-2916 ; DOI: 10.1007/s10898-012-0014-1
fulltext: fulltext
issn:
  • 1573-2916
  • 15732916
  • 0925-5001
  • 09255001
url: Link


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subjectMinimax ; Verified computations ; Fritz John equations
descriptionMinimax problems can be approached by reformulating them into smooth problems with constraints or by dealing with the non-smooth objective directly. We focus on verified enclosures of all globally optimal points of such problems. In smooth problems in branch and bound algorithms, interval Newton methods can be used to verify existence and uniqueness of solutions, to be used in eliminating regions containing such solutions, and point Newton methods can be used to obtain approximate solutions for good upper bounds on the global optimum. We analyze smooth reformulation approaches, show weaknesses in them, and compare reformulation to solving the non-smooth problem directly. In addition to analysis and illustrative problems, we exhibit the results of numerical computations on various test problems.
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titleOn smooth reformulations and direct non-smooth computations for minimax problems
descriptionMinimax problems can be approached by reformulating them into smooth problems with constraints or by dealing with the non-smooth objective directly. We focus on verified enclosures of all globally optimal points of such problems. In smooth problems in branch and bound algorithms, interval Newton methods can be used to verify existence and uniqueness of solutions, to be used in eliminating regions containing such solutions, and point Newton methods can be used to obtain approximate solutions for good upper bounds on the global optimum. We analyze smooth reformulation approaches, show weaknesses in them, and compare reformulation to solving the non-smooth problem directly. In addition to analysis and illustrative problems, we exhibit the results of numerical computations on various test problems.
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abstractMinimax problems can be approached by reformulating them into smooth problems with constraints or by dealing with the non-smooth objective directly. We focus on verified enclosures of all globally optimal points of such problems. In smooth problems in branch and bound algorithms, interval Newton methods can be used to verify existence and uniqueness of solutions, to be used in eliminating regions containing such solutions, and point Newton methods can be used to obtain approximate solutions for good upper bounds on the global optimum. We analyze smooth reformulation approaches, show weaknesses in them, and compare reformulation to solving the non-smooth problem directly. In addition to analysis and illustrative problems, we exhibit the results of numerical computations on various test problems.
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doi10.1007/s10898-012-0014-1
pages1091-1111
date2013-12