89
Views
3
CrossRef citations to date
0
Altmetric
Articles

Optimality conditions for linear copositive programming problems with isolated immobile indices

& ORCID Icon
Pages 145-164 | Received 16 Mar 2018, Accepted 11 Oct 2018, Published online: 15 Nov 2018

References

  • Motzkin TS. Copositive quadratic forms. Natl Bur Standartds Rep. 1952;1818:11–22.
  • Motzkin TS, Straus EG. Maxima for graphs and a new proof of a theorem of Turán. Can J Math. 1965;17:533–540. doi: 10.4153/CJM-1965-053-6
  • Preisig JC. Copositivity and the minimization of quadratic functions with nonnegativity and quadratic equality constraints. SIAM J Control Optim. 1996;34:1135–1150. doi: 10.1137/S0363012993251894
  • Bomze IM, Dür M, de Klerk E, et al. On copositive programming and standard quadratic optimization problems. J Glob Optim. 2000;18:301–320. doi: 10.1023/A:1026583532263
  • Hiriart-Urruty JB, Seeder A. A variational approach to copositive matrices. SIAM Rev. 2010;52:593–629. doi: 10.1137/090750391
  • Berman A, Shaked-Monderer N. Completely positive matrices. River Edge (NJ): World Scientific Publishing Co. Inc.; 2003.
  • Berman A, Dür M, Shaked-Monderer N. Open problems in the theory of completely positive and copositive matrices. Electron J Linear Algebra. 2015;29:46–58. doi: 10.13001/1081-3810.2943
  • Burer S. On the copositive representation of binary and continuous nonconvex quadratic programs. Math Program. 2009;120:479–495. doi: 10.1007/s10107-008-0223-z
  • Dür M. Copositive programming – a survey. In: Diehl M, Glineur F, Jarlebring E, Michielis W, editors. Recent advances in optimization and its applications in engineering. Berlin: Springer-Verlag; 2010. X1: p. 535.
  • Amaral P, Bomze I, Júdice J. Copositivity and constrained fractional quadratic problems. Math Program. 2014;146(1–2):325–350. doi: 10.1007/s10107-013-0690-8
  • de Klerk E, Pasechnik DV. Approximation of the stability number number of a graph via copositive programming. SIAM J Optim. 2002;12:875–892. doi: 10.1137/S1052623401383248
  • Bomze IM. Copositive optimization - recent developments and applications. EJOR. 2012;216(3):509–520. doi: 10.1016/j.ejor.2011.04.026
  • Murtu KG, Kabadi SN. Some NP-complete problems in quadratic and nonlinear programming. Math Program. 1987;39:117–129. doi: 10.1007/BF02592948
  • Bunfuss S, Dür M. Algorithmic copositivity detection by simplicial partition. Linear Algebra Appl. 2008;428:1511–1523. doi: 10.1016/j.laa.2007.09.035
  • Dickinson PJC, Hildebrand R. Considering copositivity locally. J Math Anal Appl. 2016;437:1184–1195. doi: 10.1016/j.jmaa.2016.01.063
  • Yang S-j, Li X-x. Algorithms for determining the copositivity of a given symmetric matrix. Linear Algebra Appl. 2009;430:609–618. doi: 10.1016/j.laa.2008.07.028
  • Bomze I, Schachinger W, Uchida G. Think co(mpletely)positive !-matrix properties, examples and a clustered bibliography on copositive optimization. J Global Optim. 2012;52:423–445. doi: 10.1007/s10898-011-9749-3
  • Bonnans JF, Shapiro A. Perturbation analysis of optimization problems. New-York (NY): Springer-Verlag; 2000.
  • Kostyukova OI, Tchemisova TV. Optimality conditions for convex Semi-Infinite Programming problems with finitely representable compact index sets. J Optim Theory Appl. 2017;175(1):76–103. doi: 10.1007/s10957-017-1150-z
  • Ye JJ. Constraint Qualifications and necessary optimality conditions for optimization problems with variational inequality constraints. SIAM J Optim. 2000;10(4):943–962. doi: 10.1137/S105262349834847X
  • Ramana MV. An exact duality theory for semidefinite programming and its complexity implications. New Brunswick (NJ): Rutgers University; 1995. (DIMACS Technical report 95-02R; RUTCOR).
  • Tunçel L, Wolkowicz H. Strong duality and minimal representations for cone optimization. Comput Optim Appl. 2013;53:619–648. doi: 10.1007/s10589-012-9480-0
  • Kostyukova OI, Tchemisova TV. Implicit optimality criterion for convex SIP problem with box constrained index set. TOP. 2012;20(2):475–502. doi: 10.1007/s11750-011-0189-5
  • Kostyukova OI, Tchemisova TV. A constructive algorithm for determination of immobile indices in convex SIP problems with polyhedral index sets. Working paper, University of Aveiro 2012. Available from: http://hdl.handle.net/10773/8888.
  • Kostyukova OI, Tchemisova TV. Optimality criteria without constraint qualification for linear semidefinite problems. special issue “Algebraic Techniques in Graph Theory and Optimisation”. JMS- J Math Sci. 2012;182(2):126–143. doi: 10.1007/s10958-012-0734-2
  • Kostyukova OI, Tchemisova TV. On a constructive approach to optimality conditions for convex SIP problems with polyhedral index sets. Optimization. 2014;63(1):67–91. doi: 10.1080/02331934.2013.853062
  • Ahmed F, Dür M, Still G. Copositive programming via semi-infinite optimization. J Optim Theory Appl. 2013;159:322–340. doi: 10.1007/s10957-013-0344-2
  • Berman A, Shaked-Monderer N. Completely positive matrices. Singapore: World Scientific; 2003.
  • Goberna MA, López MA. Linear semi-infinite optimization. Chichester: Wiley; 1998.
  • Shapiro A. Duality, optimality conditions and perturbation analysis. In: Saigal R, Vandenberghe L, Wolkowicz H, editors. Semidefinite programming and applications handbook. Boston (MA): Kluwer Academic Publishers; 2000. p. 67–92.
  • Borwein JM, Wolkowicz H. Characterization of optimality for the abstract convex program with finite-dimensional range. J Aust Math Soc, Ser A. 1980/1981;30(4):390–411. doi: 10.1017/S1446788700017882

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.