Subdivision of simplices relative to a cutting plane and finite concave minimization
Identifieur interne : 001254 ( Istex/Corpus ); précédent : 001253; suivant : 001255Subdivision of simplices relative to a cutting plane and finite concave minimization
Auteurs : Michael NastSource :
- Journal of Global Optimization [ 0925-5001 ] ; 1996-07-01.
Abstract
Abstract: In this article, our primary concern is the classical problem of minimizing globally a concave function over a compact polyhedron (Problem (P)). We present a new simplicial branch and bound approach, which combines triangulations of intersections of simplices with halfspaces and ideas from outer approximation in such a way, that a class of finite algorithms for solving (P) results. For arbitrary compact convex feasible sets one obtains a not necessarily finite but convergent algorithm. Theoretical investigations include determination of the number of simplices in each applied triangulation step and bounds on the number of iterations in the resulting algorithms. Preliminary numerical results are given, and additional applications are sketched.
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DOI: 10.1007/BF00121751
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We present a new simplicial branch and bound approach, which combines triangulations of intersections of simplices with halfspaces and ideas from outer approximation in such a way, that a class of finite algorithms for solving (P) results. For arbitrary compact convex feasible sets one obtains a not necessarily finite but convergent algorithm. Theoretical investigations include determination of the number of simplices in each applied triangulation step and bounds on the number of iterations in the resulting algorithms. Preliminary numerical results are given, and additional applications are sketched.</div></front></TEI><istex><corpusName>springer</corpusName><author><json:item><name>Michael Nast</name><affiliations><json:string>Fachbereich IV, Department of Mathematics, University of Trier, Trier, Germany</json:string><json:string>E-mail: nast@uni-trier.de</json:string></affiliations></json:item></author><articleId><json:string>BF00121751</json:string><json:string>Art4</json:string></articleId><language><json:string>eng</json:string></language><originalGenre><json:string>OriginalPaper</json:string></originalGenre><abstract>Abstract: In this article, our primary concern is the classical problem of minimizing globally a concave function over a compact polyhedron (Problem (P)). We present a new simplicial branch and bound approach, which combines triangulations of intersections of simplices with halfspaces and ideas from outer approximation in such a way, that a class of finite algorithms for solving (P) results. For arbitrary compact convex feasible sets one obtains a not necessarily finite but convergent algorithm. Theoretical investigations include determination of the number of simplices in each applied triangulation step and bounds on the number of iterations in the resulting algorithms. Preliminary numerical results are given, and additional applications are sketched.</abstract><qualityIndicators><score>6.32</score><pdfVersion>1.3</pdfVersion><pdfPageSize>450 x 677 pts</pdfPageSize><refBibsNative>false</refBibsNative><keywordCount>0</keywordCount><abstractCharCount>762</abstractCharCount><pdfWordCount>11991</pdfWordCount><pdfCharCount>49530</pdfCharCount><pdfPageCount>29</pdfPageCount><abstractWordCount>110</abstractWordCount></qualityIndicators><title>Subdivision of simplices relative to a cutting plane and finite concave minimization</title><refBibs><json:item><host><pages><last>177</last><first>165</first></pages><author><json:item><name>H,P Benson</name></json:item></author><title>A Finite Algorithm for Concave Minimization over a Polyhedron</title><publicationDate>1985</publicationDate></host></json:item><json:item><author><json:item><name>H,P Benson</name></json:item></author><host><pages><last>148</last><first>43</first></pages><author></author><title>Concave 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Jacobsen</name></json:item></author><host><volume>13</volume><pages><last>487</last><first>479</first></pages><author></author><title>Mathematics of Operations Research</title><publicationDate>1988</publicationDate></host><title>Exhaustive Nondegenerate Conical Process for Concave Minimization on Convex Polytopes</title><publicationDate>1988</publicationDate></json:item><json:item><author><json:item><name>R Horst</name></json:item></author><host><volume>10</volume><pages><last>321</last><first>312</first></pages><author></author><title>Mathematical Programming</title><publicationDate>1976</publicationDate></host><title>An Algorithm for Nonconvex Programming Problems</title><publicationDate>1976</publicationDate></json:item><json:item><host><pages><last>289</last><first>271</first></pages><author><json:item><name>R Horst</name></json:item><json:item><name>P,M Pardalos</name></json:item><json:item><name>N,V Thoai</name></json:item></author><title>Introduction to Global Optimization Modification, Implementation and Comparison of Three Algorithms for Globally solving Concave Minimization Problems, Computing</title><publicationDate>1988</publicationDate></host></json:item><json:item><author><json:item><name>R Horst</name></json:item><json:item><name>H Tuy</name></json:item></author><host><author></author><title>Global Optimization (Deterministic Approaches)</title><publicationDate>1993</publicationDate></host><publicationDate>1993</publicationDate></json:item><json:item><author><json:item><name>R,B Hughes</name></json:item></author><host><volume>118</volume><pages><last>118</last><first>75</first></pages><author></author><title>Discrete Mathematics</title><publicationDate>1993</publicationDate></host><title>Minimum-eardinality triangulations of the d--cube for d = 5 and d = 6</title><publicationDate>1993</publicationDate></json:item><json:item><author><json:item><name>R,B Hughes</name></json:item><json:item><name>M,R Anderson</name></json:item></author><host><pages><last>256</last><first>253</first></pages><author></author><title>A triangulation of the 6-cube with 308 simplices, Discrete Mathematics</title><publicationDate>1993</publicationDate></host><publicationDate>1993</publicationDate></json:item><json:item><author><json:item><name>C,H Papadimitriou</name></json:item><json:item><name>K Steiglitz</name></json:item></author><host><pages><last>382</last><first>373</first></pages><author></author><title>Combinatorial Optimization: .Algorithms and Complexity Optimal Facility Location with Concave Costs</title><publicationDate>1974</publicationDate></host><publicationDate>1974</publicationDate></json:item><json:item><author><json:item><name>B,T Tam</name></json:item><json:item><name>V,T Ban</name></json:item><json:item><name> Russian</name></json:item><json:item><name>M,J Todd</name></json:item></author><host><pages><last>714</last><first>709</first></pages><author></author><title>Effect of the Subdivision Strategy on Convergence and Efficiency of Some Global Optimization Algorithms</title><publicationDate>1976</publicationDate></host><title>Minimization of a Concave Function Under Linear Constraints The Computation of Fixed Points and Applications</title><publicationDate>1976</publicationDate></json:item></refBibs><genre><json:string>research-article</json:string></genre><host><volume>9</volume><pages><last>93</last><first>65</first></pages><issn><json:string>0925-5001</json:string></issn><issue>1</issue><subject><json:item><value>Computer Science, general</value></json:item><json:item><value>Real Functions</value></json:item><json:item><value>Optimization</value></json:item><json:item><value>Operation Research/Decision 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We present a new simplicial branch and bound approach, which combines triangulations of intersections of simplices with halfspaces and ideas from outer approximation in such a way, that a class of finite algorithms for solving (<Emphasis Type="Italic">P</Emphasis>) results. For arbitrary compact convex feasible sets one obtains a not necessarily finite but convergent algorithm. Theoretical investigations include determination of the number of simplices in each applied triangulation step and bounds on the number of iterations in the resulting algorithms. 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