On solving a non-convex quadratic programming problem involving resistance distances in graphs

Quadratic programming problems involving distance matrix ( D ) that arises in trees are considered in the literature by Dankelmann (Discrete Math 312:12–20, 2012 ), Bapat and Neogy (Ann Oper Res 243:365–373, 2016 ). In this paper, we consider the question of solving the quadratic programming problem...

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Published in:	Annals of operations research Vol. 287; no. 2; pp. 643 - 651
Main Authors:	Dubey, Dipti, Neogy, S. K.
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.04.2020 Springer Springer Nature B.V
Subjects:	Analysis Business and Management Combinatorics Distance matrices Graphic methods Graphs Mathematical programming Matrices (mathematics) Methods Operations research Operations Research/Decision Theory Problem solving Quadratic programming S.I.: Game theory and optimization Studies Theory of Computation India Non-convex quadratic programming Laplacian matrix Polynomial time algorithm Symmetric bimatrix game Resistance distance 90C33
ISSN:	0254-5330, 1572-9338
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Abstract	Quadratic programming problems involving distance matrix ( D ) that arises in trees are considered in the literature by Dankelmann (Discrete Math 312:12–20, 2012 ), Bapat and Neogy (Ann Oper Res 243:365–373, 2016 ). In this paper, we consider the question of solving the quadratic programming problem of finding maximum of x T R x subject to x being a nonnegative vector with sum 1 and show that for the class of simple graphs with resistance distance matrix ( R ) which are not necessarily a tree, this problem can be reformulated as a strictly convex quadratic programming problem. An application to symmetric bimatrix game is also presented.
AbstractList	Quadratic programming problems involving distance matrix (D) that arises in trees are considered in the literature by Dankelmann (Discrete Math 312:12–20, 2012), Bapat and Neogy (Ann Oper Res 243:365–373, 2016). In this paper, we consider the question of solving the quadratic programming problem of finding maximum of xTRx subject to x being a nonnegative vector with sum 1 and show that for the class of simple graphs with resistance distance matrix (R) which are not necessarily a tree, this problem can be reformulated as a strictly convex quadratic programming problem. An application to symmetric bimatrix game is also presented. Quadratic programming problems involving distance matrix ( D ) that arises in trees are considered in the literature by Dankelmann (Discrete Math 312:12–20, 2012 ), Bapat and Neogy (Ann Oper Res 243:365–373, 2016 ). In this paper, we consider the question of solving the quadratic programming problem of finding maximum of x T R x subject to x being a nonnegative vector with sum 1 and show that for the class of simple graphs with resistance distance matrix ( R ) which are not necessarily a tree, this problem can be reformulated as a strictly convex quadratic programming problem. An application to symmetric bimatrix game is also presented. Quadratic programming problems involving distance matrix (D) that arises in trees are considered in the literature by Dankelmann (Discrete Math 312:12-20, 2012 (See CR11)), Bapat and Neogy (Ann Oper Res 243:365-373, 2016 (See CR6)). In this paper, we consider the question of solving the quadratic programming problem of finding maximum of [Formula omitted] subject to x being a nonnegative vector with sum 1 and show that for the class of simple graphs with resistance distance matrix (R) which are not necessarily a tree, this problem can be reformulated as a strictly convex quadratic programming problem. An application to symmetric bimatrix game is also presented.
Audience	Academic
Author	Neogy, S. K. Dubey, Dipti
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Keywords	Non-convex quadratic programming Laplacian matrix Polynomial time algorithm Symmetric bimatrix game Resistance distance 90C33
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References_xml	– reference: TardosEA strongly polynomial algorithm to solve combinatorial linear programsOperations Research19863425025610.1287/opre.34.2.250 – reference: XiaoWGutmanIOn resistance matricesMATCH Communications in Mathematical and in Computer Chemistry2003496781 – reference: LemkeCEBimatrix equilibrium points and mathematical programmingManagement Science19651168168910.1287/mnsc.11.7.681 – reference: DankelmannPAverage distance in weighted graphsDiscrete Mathematics2012312122010.1016/j.disc.2011.02.010 – reference: BapatRBThe Laplacian matrix of a graphMathematics Student199665214223 – reference: BapatRBResistance distance in graphsMathematics Student1999688798 – reference: BapatRBResistance matrix of a weighted graphMATCH-Communications in Mathematical and in Computer Chemistry2004507382 – reference: GranotFSkorin-KapovJTowards a strongly polynomial algorithm for strictly convex quadratic programs: An extension of Tardos’ algorithmMathematical Programming19904622523610.1007/BF01585740 – reference: BomzeIMLocatelliMTardellaFNew and old bounds for standard quadratic optimization: Dominance, equivalence and incomparabilityMathematical Programming2008115316410.1007/s10107-007-0138-0 – reference: BomzeIMRegularity versus degeneracy in dynamics, games and optimization: A unified approach to different aspectsSIAM Review20024439441410.1137/S00361445003756 – reference: XiaoWGutmanIResistance distance and Laplacian spectrumTheoretical Chemistry Accounts200311028428910.1007/s00214-003-0460-4 – reference: BapatRBNeogySKOn a quadratic programming problem involving distances in treesAnnals of Operations Research201624336537310.1007/s10479-014-1743-y – reference: CottleRWPangJSStoneREThe linear complementarity problem2012New YorkAcademic Press – reference: ScozzariATardellaFA clique algorithm for standard quadratic programmingDiscrete Applied Mathematics20081562439244810.1016/j.dam.2007.09.020 – reference: EstradaEThe structure of complex networks: Theory and applications2011New YorkOxford University Press10.1093/acprof:oso/9780199591756.001.0001 – reference: KojimaMMizunoSYoshiseAA polynomial-time algorithm for a class of linear complementarity problemsMathematical Programming19894412610.1007/BF01587074 – reference: KleinDRandićMResistance distanceJournal of Mathematical Chemistry199312819510.1007/BF01164627 – reference: BapatRBGraphs and matrices2010LondonUniversitext, Springer10.1007/978-1-84882-981-7 – reference: BapatRBGutmanaIXiaoWA simple method for computing resistance distanceZeitschrift für Naturforschung A20035849449810.1515/zna-2003-9-1003 – reference: HjorthPLisoněkPMarkvorsenSThomassenCFinite metric spaces of strictly negative typeLinear Algebra and its Applications199827025527310.1016/S0024-3795(97)00242-5 – reference: BapatRBSivasubramanianSIdentities for minors of the Laplacian, resistance and distance matricesLinear Algebra and its Applications20114351479148910.1016/j.laa.2011.03.028 – reference: MangasarianOLStoneHTwo-person nonzero-sum games and quadratic programmingJournal of Mathematical Analysis and Applications1964934835510.1016/0022-247X(64)90021-6 – volume-title: Graphs and matrices year: 2010 ident: 3018_CR1 doi: 10.1007/978-1-84882-981-7 – volume: 68 start-page: 87 year: 1999 ident: 3018_CR2 publication-title: Mathematics Student – volume-title: The structure of complex networks: Theory and applications year: 2011 ident: 3018_CR12 doi: 10.1093/acprof:oso/9780199591756.001.0001 – volume: 110 start-page: 284 year: 2003 ident: 3018_CR21 publication-title: Theoretical Chemistry Accounts doi: 10.1007/s00214-003-0460-4 – volume: 435 start-page: 1479 year: 2011 ident: 3018_CR7 publication-title: Linear Algebra and its Applications doi: 10.1016/j.laa.2011.03.028 – volume: 115 start-page: 31 year: 2008 ident: 3018_CR9 publication-title: Mathematical Programming doi: 10.1007/s10107-007-0138-0 – volume: 65 start-page: 214 year: 1996 ident: 3018_CR3 publication-title: Mathematics 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Snippet	Quadratic programming problems involving distance matrix ( D ) that arises in trees are considered in the literature by Dankelmann (Discrete Math 312:12–20,... Quadratic programming problems involving distance matrix (D) that arises in trees are considered in the literature by Dankelmann (Discrete Math 312:12-20, 2012... Quadratic programming problems involving distance matrix (D) that arises in trees are considered in the literature by Dankelmann (Discrete Math 312:12–20,...
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SubjectTerms	Analysis Business and Management Combinatorics Distance matrices Graphic methods Graphs Mathematical programming Matrices (mathematics) Methods Operations research Operations Research/Decision Theory Problem solving Quadratic programming S.I.: Game theory and optimization Studies Theory of Computation
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Title	On solving a non-convex quadratic programming problem involving resistance distances in graphs
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