Copositive optimization - Recent developments and applications

European Journal of Operational Research

Volumn 216, Issue 3, 2012, Pages 509-520

  Bomze, Immanuel M ^a  

^a UNIVERSITY OF VIENNA   (Austria)

Author keywords

Clique number; Completely positive matrix; Convexity gap; Crossing number; Robust optimization; Standard quadratic optimization

Indexed keywords

COMBINATORIAL OPTIMIZATION; QUADRATIC PROGRAMMING; STOCHASTIC SYSTEMS;

CLIQUE NUMBER; COMPLETELY POSITIVE MATRIX; CONVEXITY GAP; CROSSING NUMBER; ROBUST OPTIMIZATION; STANDARD QUADRATIC OPTIMIZATION;

STOCHASTIC MODELS;

EID: 80053917198     PISSN: 03772217     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.ejor.2011.04.026     Document Type: Article

References (128)

1. P.A. Amaral, I.M. Bomze, J.J. Júdice, Copositivity and constrained fractional quadratic problems. Technical Report 2010-05, ISDS, Univ. Vienna. Available from: , 2010.
2. M. Anitescu, J.F. Cremer, and F.A. Potra On the existence of solutions to complementarity formulations of contact problems with friction Complementarity and Variational Problems (Baltimore, MD, 1995) 1997 SIAM Philadelphia, PA 12 21
3. M. Anitescu, and F.A. Potra A time-stepping method for stiff multibody dynamics with contact and friction International Journal for Numerical Methods in Engineering 55 2002 753 784 (Pubitemid 35157165)
4. K.M. Anstreicher, and S. Burer Computable representations for convex hulls of low-dimensional quadratic forms Mathematical Programming 124 2010 33 43
5. D. Azé, and J.-B. Hiriart-Urruty Optimal Hoffman-type estimates in eigenvalue and semidefinite inequality constraints Journal of Global Optimization 24 2002 133 147
6. M. Bellare, and P. Rogaway The complexity of approximating a nonlinear program Mathematical Programming 69 1995 429 441
7. A. Berman, and N. Shaked-Monderer Completely Positive Matrices 2003 World Scientific River Edge, NJ
8. I.M. Bomze Remarks on the recursive structure of copositivity Journal of Information and Optimization Sciences 8 1987 243 260
9. I.M. Bomze Copositivity conditions for global optimality in indefinite quadratic programming problems Czechosl Journal of Operations Research 1 1992 7 19
10. I.M. Bomze Detecting all evolutionarily stable strategies Journal of Optimization Theory and Applications 75 1992 313 329 (Pubitemid 23581291)
11. I.M. Bomze Block pivoting and shortcut strategies for detecting copositivity Linear Algebra and its Applications 248 1996 161 184
12. I.M. Bomze On standard quadratic optimization problems Journal of Global Optimization 13 1998 369 387 (Pubitemid 128507399)
13. I.M. Bomze Linear-time copositivity detection for tridiagonal matrices and extension to block-tridiagonality SIAM Journal on Matrix Analysis and Applications 21 2000 840 848
14. I.M. Bomze Regularity versus degeneracy in dynamics, games, and optimization: A unified approach to different aspects SIAM Review 44 2002 394 414
15. I.M. Bomze Perron-Frobenius property of copositive matrices, and a block copositivity criterion Linear Algebra and its Applications 429 2008 68 71
16. I.M. Bomze Copositive optimization C. Floudas, P. Pardalos, Encyclopedia of Optimization 2009 Springer New York 561 564
17. I.M. Bomze Global optimization: A quadratic programming perspective Nonlinear Optimization Lecture Notes in Math vol. 1989 2010 Springer Berlin 1 53
18. I.M. Bomze, and E. de Klerk Solving standard quadratic optimization problems via linear, semidefinite and copositive programming Journal of Global Optimization 24 2002 163 185
19. I.M. Bomze, M. Dür, E. de Klerk, C. Roos, A.J. Quist, and T. Terlaky On copositive programming and standard quadratic optimization problems Journal of Global Optimization 18 2000 301 320
20. I.M. Bomze, G. Eichfelder, Copositivity detection by difference-of-convex decomposition and ω-subdivision. Technical Report 2010-01, ISDS, Univ. Vienna. Available from: , 2010.
21. I.M. Bomze, F. Frommlet, and M. Locatelli Copositivity cuts for improving SDP bounds on the clique number Mathematical Programming 124 2010 13 32
22. I.M. Bomze, and F. Jarre A note on Burer's copositive representation of mixed-binary QPs Optimization Letters 4 2010 465 472
23. I.M. Bomze, F. Jarre, and F. Rendl Quadratic factorization heuristics for copositive programming Mathematical Programming Computation 3 2011 37 57
24. I.M. Bomze, and C. Lemaréchal Necessary conditions for local optimality in difference-of-convex programming Journal of Convex Analysis 17 2010 673 680
25. I.M. Bomze, and W. Schachinger Multi-standard quadratic optimization: interior point methods and cone programming reformulation Computational Optimization and Applications 45 2010 237 256
26. I.M. Bomze, W. Schachinger, G. Uchida, Think co(mpletely) positive! - properties, examples and a commented bibliography on copositive optimization, Journal of Global Optimization, in press.
27. J.M. Borwein Necessary and sufficient conditions for quadratic minimality Numerical Functional Analysis and Optimization 5 1982 127 140 (Pubitemid 13590972)
28. B. Brogliato, A. ten Dam, L. Paoli, F. Génot, and M. Abadie Numerical simulation of finite dimensional multibody nonsmooth mechanical systems ASME Applied Mechanics Reviews 55 2002 107 150
29. M. Buliga Four applications of majorization to convexity in the calculus of variations Linear Algebra and its Applications 429 2008 1528 1545
30. S. Bundfuss, and M. Dür Algorithmic copositivity detection by simplicial partition Linear Algebra and its Applications 428 2008 1511 1523 (Pubitemid 351191953)
31. S. Bundfuss, and M. Dür An adaptive linear approximation algorithm for copositive programs SIAM Journal on Optimization 20 2009 30 53
32. S. Bundfuss, and M. Dür Copositive Lyapunov functions for switched systems over cones Systems & Control Letters 58 2009 342 345
33. S. Burer On the copositive representation of binary and continuous nonconvex quadratic programs Mathematical Programming 120 2009 479 495
34. S. Burer, Copositive programming. In: M.F. Anjos, J.B. Lasserre, (Eds.), Handbook of Semidefinite, Cone and Polynomial Optimization: Theory, Algorithms, Software and Applications, International Series in Operations Research and Management Science, Springer, New York, in press.
35. S. Burer, K.M. Anstreicher, and M. Dür The difference between 5 × 5 doubly nonnegative and completely positive matrices Linear Algebra and its Applications 431 2009 1539 1552
36. S. Burer, H. Dong, Separation and Relaxation for cones of quadratic forms, Working paper, Department of Management Sciences, University of Iowa, Iowa City IA. Available from: , 2010.
37. S. Busygin Copositive programming C. Floudas, P. Pardalos, Encyclopedia of Optimization 2009 Springer New York 564 567
38. M.K. Camlibel, and J.M. Schumacher Copositive Lyapunov functions V. Blondel, A. Megretski, Unsolved Problems in Mathematical Systems and Control Theory 2004 Princeton University Press Princeton NJ 189 193 Available from:
39. T. Coleman, and J. Liu An interior Newton method for quadratic programming Mathematical Programming 85 1999 491 523
40. L.B. Contesse Une caractérisation complte des minima locaux Numerical Mathematics 34 1980 315 332
41. R.W. Cottle, G.J. Habetler, and C.E. Lemke Quadratic forms semi-definite over convex cones Proceedings of the Princeton Symposium on Mathematical Programming (Princeton Univ.; 1967) 1970 Princeton Univ. Press Princeton, N.J 551 565
42. B. Dacorogna Necessary and sufficient conditions for strong ellipticity of isotropic functions in any dimension Discrete and Continuous Dynamic Systems-Series B 1 2001 257 263 (Pubitemid 33771908)
43. G. Danninger, A recursive algorithm for determining (strict) copositivity of a symmetric matrix, in: XIV Symposium on Operations Research (Ulm, 1989), Frankfurt am Main: Hain, volume 62 of Methods Oper. Res.; 1990, pp. 45-52.
44. G. Danninger, and I.M. Bomze Using copositivity for global optimality criteria in concave quadratic programming problems Mathematical Programming 62 1993 575 580
45. E. de Klerk The complexity of optimizing over a simplex, hypercube or sphere: A short survey Central European Journal of Operations Research 16 2008 111 125
46. E. de Klerk, C. Dobre, D.V. Pasechnik, Numerical block diagonalization of matrix -algebras with application to semidefinite programming, Mathematical Programming, in press.
47. n SIAM Journal on Discrete Mathematics 20 2006 189 202 (Pubitemid 46352095)
48. E. de Klerk, and D.V. Pasechnik Approximation of the stability number of a graph via copositive programming SIAM Journal on Optimization 12 2002 875 892
49. E. de Klerk, and D.V. Pasechnik A linear programming reformulation of the standard quadratic optimization problem Journal of Global Optimization 37 2007 75 84 (Pubitemid 44876768)
50. E. de Klerk, D.V. Pasechnik, and A. Schrijver Reduction of symmetric semidefinite programs using the regular *-representation Mathematical Programming Series B 109 2007 613 624 (Pubitemid 46197869)
51. P.H. Diananda On non-negative forms in real variables some or all of which are non-negative Proceedings of the Cambridge Philosophical Society 58 1962 17 25
52. C. Dobre, Semidefinite programming approaches for structured combinatorial optimization problems. Ph.d. dissertation, Univ. Tilburg, The Netherlands, 2011.
53. H. Dong, Symmetric tensor approximation hierarchies for the completely positive cone. Working paper, Department of Management Sciences, University of Iowa, Iowa City, IA. Available from: , 2010.
54. H. Dong, and K. Anstreicher A note on "5 × 5 completely positive matrices" Linear Algebra and its Application 433 2010 1001 1004
55. H. Dong, K. Anstreicher, Separating Doubly Nonnegative and Completely Positive Matrices. Working paper, Department of Management Sciences, University of Iowa, Iowa City, IA. Available from: , 2010.
56. I. Dukanovic, and F. Rendl Copositive programming motivated bounds on the stability and the chromatic numbers Mathematical Programming 121 2010 249 268
57. M. Dür Copositive programming - A survey M. Diehl, F. Glineur, E. Jarlebring, W. Michiels, Recent Advances in Optimization and its Applications in Engineering 2010 Springer Berlin Heidelberg New York 3 20
58. G. Eichfelder, and J. Jahn Set-semidefinite optimization Journal of Convex Analysis 15 2008 767 801
59. G. Eichfelder, J. Povh, On reformulations of nonconvex quadratic programs over convex cones by set-semidefinite constraints. Preprint 342 Institut fuer Angewandte Mathematik, Martensstraße 3, D-91058 Erlangen. Available from: , 2010.
60. A. Engau, M.F. Anjos, I.M. Bomze, Cutting planes and copositive programming for stable set. Technical Report 2011-04, ISOR, Univ. Vienna, 2011.
61. F. Facchinei, and J.-S. Pang Finite-dimensional variational inequalities and complementarity problems Springer Series in Operations Research vol. I 2003 Springer-Verlag New York
62. F. Facchinei, and J.-S. Pang Finite-dimensional variational inequalities and complementarity problems Springer Series in Operations Research vol. II 2003 Springer-Verlag New York
63. R. Fletcher Practical Methods of Optimization Constrained Optimization vol. 2 1981 Wiley New York
64. P.E. Gill, W. Murray, and M.H. Wright Practical Optimization 1981 Academic Press London
65. D. Gourion, and A. Seeger Critical angles in polyhedral convex cones: Numerical and statistical considerations Mathematical Programming 123 2010 173 198
66. E. Guslitser, Uncertatinty-immunized solutions in linear programming. Master thesis, Technion, Israeli Institute of Technology, IE&M faculty, 2002.
67. N. Gvozdenović, and M. Laurent The operator Ψ for the chromatic number of a graph SIAM Journal on Optimization 19 2008 572 591
68. K.P. Hadeler On copositive matrices Linear Algebra and its Applications 49 1983 79 89
69. P. Hadjicostas Copositive matrices and Simpson's paradox Linear Algebra and its Applications 264 1997 475 488
70. R.H.R. Hahnloser, H.S. Seung, and J.-J.E. Slotine Permitted and forbidden sets in symmetric threshold-linear networks Neural Comput. 15 2003 621 638 (Pubitemid 37049819)
71. M. Hall Jr.; and M. Newman Copositive and completely positive quadratic forms Proceedings of the Cambridge Philosophical Society 59 1963 329 339
72. 1- Acta Math. 182 1999 105 142
73. E. Haynsworth, and A.J. Hoffman Two remarks on copositive matrices Linear Algebra and its Application 2 1969 387 392
74. J.-B. Hiriart-Urruty, and A. Seeger A variational approach to copositive matrices SIAM Review 52 2010 593 629
75. C. Humes Jr.; J. Ou, and P.R. Kumar The delay of open Markovian queueing networks: Uniform functional bounds, heavy traffic pole multiplicities, and stability Mathematics of Operations Research 22 1997 921 954 (Pubitemid 127653367)
76. K.D. Ikramov An algorithm, linear with respect to time, for verifying the copositivity of an acyclic matrix Computational Mathematics and Mathematical Physics 42 2002 1701 1703 (Pubitemid 36137070)
77. K.D. Ikramov, and N.V. Savel'eva Conditionally definite matrices Journal of Mathematical Sciences 98 2000 1 50
78. A. Iusem, and A. Seeger Searching for critical angles in a convex cone Mathematical Programming 120 2009 3 25
79. D.H. Jacobson, Extensions of linear-quadratic control, optimization and matrix theory. London: Academic Press [Harcourt Brace Jovanovich Publishers], Mathematics in Science and Engineering, vol. 133, 1977.
80. F. Jarre, Burer's key assumption for semidefinite and doubly nonnegative relaxations. Optimization Letters. Available from: , to appear.
81. C.R. Johnson, and R. Reams Spectral theory of copositive matrices Linear Algebra and its Applications 395 2005 275 281
82. J.J. Júdice, H.D. Sherali, and I.M. Ribeiro The eigenvalue complementarity problem Computational Optimization and Applications 37 2007 139 156 (Pubitemid 46709792)
83. P.R. Kumar, and S.P. Meyn Duality and linear programs for stability and performance analysis of queueing networks and scheduling policies IEEE Transactions on Automatic Control 41 1996 4 17 (Pubitemid 126768435)
84. Y. Kwon On Hadamard stability for compressible viscoelastic constitutive equations Journal of Non-Newtonian Fluid Mechanics 65 1996 151 163 (Pubitemid 126394849)
85. C. Lacoursire, Splitting methods for dry frictional contact problems in rigid multibody systems: Preliminary performance results, in: Conference Proceedings from SIGRAD2003, Umeå, Sweden, 2003, pp. 20-21.
86. J.B. Lasserre Global optimization with polynomials and the problem of moments SIAM Journal on Optimization 11 2000/01 796 817
87. M. Laurent, and F. Rendl Semidefinite programming and integer programming K. Aardal, G. Nemhauser, R. Weismantel, Handbook on Discrete Optimization 2005 Elsevier Amsterdam 393 514
88. E. Leonardi, M. Mellia, M. Ajmone Marsan, and F. Neri On the throughput achievable by isolated and interconnected input-queueing switches under multiclass traffic INFOCOM 2002 Proceedings of the 21st Annual Joint Conference of the IEEE Computer and Communications Societies vol. 3 2002 IEEE 1605 1614
89. J.-Q. Liu, T.-T. Song, and D.-Z. Du On the necessary and sufficient condition of the local optimal solution of quadratic programming Chinese Annals of Mathematics 3 1982 625 630 (English abstract)
90. M. Locatelli, F. Schoen, On convex envelopes and underestimators for bivariate functions. Available from: , 2010.
91. L. Lovász On the Shannon capacity of a graph IEEE Transactions on Information Theory IT-25 1979 1 7
92. L. Lovász, and A. Schrijver Cones of matrices and set-functions and 0-1 optimization SIAM Journal on Optimization 1 1991 166 190
93. A. Majthay Optimality conditions for quadratic programming Mathematical Programming 1 1971 359 365
94. D.H. Martin, and D.H. Jacobson Copositive matrices and definiteness of quadratic forms subject to homogeneous linear inequality constraints Linear Algebra and its Applications 35 1981 227 258
95. N. Matsubayashi, and H. Nishino An application of Lemke's method to a class of Markov decision problems European Journal of Operational Research 116 1999 584 590
96. R. McEliece, E. Rodemich, and H. Rumsey The Lovász' bound and some generalizations Journal of Combinatorics, Information & System Sciences 3 1978 134 152
97. M. Mesbahi, M.G. Safonov, and G.P. Papavassilopoulos Bilinearity and complementarity in robust control Advances in Linear Matrix Inequality Methods in Control Advances in Design and Control vol. 2 2000 SIAM Philadelphia, PA 269 292
98. J.R. Morrison, and P.R. Kumar New linear program performance bounds for queueing networks Journal of Optimization Theory and Applications 100 1999 575 597
99. J.R. Morrison, and P.R. Kumar New linear program performance bounds for closed queueing networks Discrete Event Dynamics Systems 11 2001 291 317 (Pubitemid 32839515)
100. T.S. Motzkin, and E.G. Straus Maxima for graphs and a new proof of a theorem of Turán Canadian Journal of Mathematics 17 1965 533 540
101. K.G. Murty Linear complementarity, linear and nonlinear programming Sigma Series in Applied Mathematics vol. 3 1988 Heldermann Verlag Berlin
102. K.G. Murty, and S.N. Kabadi Some NP-complete problems in quadratic and nonlinear programming Mathematical Programming 39 1987 117 129
103. E. Nadler Nonnegativity of bivariate quadratic functions on a triangle Computer Aided Geometric Design 9 1992 195 205
104. m,n Electronic Journal of Combinatorics 10 2003
105. K. Natarajan, C.P. Teo, Z. Zheng, Mixed zero-one linear programs under objective uncertainty: a completely positive representation. Operations Research. Available from: , in press.
106. National Bureau of Standards Report 1818 (1952). Projects and Publications of the National Applied Mathematics Laboratories, pp. 11-12. Quarterly Report, April through June 1952.
107. A. Ozdaglar, and P. Tseng Existence of global minima for constrained optimization Journal of Optimization Theory and Applications 128 2006 523 546 (Pubitemid 44631550)
108. P.A. Parrilo Semidefinite programming relaxations for semialgebraic problems Mathematical Programming 96 2003 293 320 (Pubitemid 44757842)
109. J. Peña, J. Vera, and L.F. Zuluaga Computing the stability number of a graph via linear and semidefinite programming SIAM Journal on Optimization 18 2007 87 105 (Pubitemid 351271322)
110. A. Pinto da Costa, and A. Seeger Cone-constrained eigenvalue problems: Theory and algorithms Computational Optimization and Applications 45 2010 25 57
111. J. Povh, and F. Rendl A copositive programming approach to graph partitioning SIAM Journal on Optimization 18 2007 223 241 (Pubitemid 351271329)
112. J. Povh, and F. Rendl Copositive and semidefinite relaxations of the quadratic assignment problem Discrete Optimization 6 2009 231 241
113. J.C. Preisig Copositivity and the minimization of quadratic functions with nonnegativity and quadratic equality constraints SIAM Journal on Control and Optimization 34 1996 1135 1150 (Pubitemid 126562706)
114. M. Putinar Positive polynomials on compact semi-algebraic sets Indiana University Mathematics Journal 42 1993 969 984
115. R.X. Qian, and C.L. DeMarco An approach to robust stability of matrix polytopes through copositive homogeneous polynomials IEEE Transactions on Automatic Control 37 1992 848 852
116. A.J. Quist, E. de Klerk, C. Roos, and T. Terlaky Copositive relaxation for general quadratic programming Optimization Methods and Software 9 1998 185 208 (Pubitemid 128542196)
117. W. Schachinger, and I. Bomze A conic duality Frank-Wolfe-type theorem via exact penalization in quadratic optimization Mathematics of Operations Research 34 2009 83 91
118. K. Schmüdgen The K-moment problem for compact semi-algebraic sets Annals of Mathematics 289 1991 203 206
119. A. Schrijver A comparison of the Delsarte and Lovasz bounds IEEE Transactions on Information Theory IT-25 1979 425 429
120. G. Stengle A Nullstellensatz and a Positivstellensatz in semialgebraic geometry Annals of Mathematics 207 1973 87 97
121. M. Tawarmalani, and N.V. Sahinidis Global optimization of mixed-integer nonlinear programs: a theoretical and computational study Mathematical Programming 99 2004 563 591 (Pubitemid 40871718)
122. P. Turán A note of welcome Journal of Graph Theory 1 1977 7 9
123. H. Väliaho Testing the definiteness of matrices on polyhedral cones Linear Algebra and its Applications 101 1988 135 165
124. L. Vandenberghe, S. Boyd, and C. Comanor Generalized Chebychev bounds via semidefinite programming SIAM Review 49 2007 52 64 (Pubitemid 46351351)
125. V.A. Yakubovich, S-procedure in nonlinear control theory, Vestnik Leningradskogo Universiteta, Ser. Matematika, Series 1, vol. 13, 1997, pp. 62-77. English translation in Vestnik Leningrad Univ.; 1977, vol. 4, pp. 73-93.
126. E.A. Yildirim, On the accuracy of uniform polyhedral approximations of the copositive cone, Optimization Methods and Software. Available from: , in press.
127. K. Zarankiewicz On a problem of P. Turán concerning graphs Fundamenta Mathematicae 41 1954 137 145
128. J. Zilinskas, M. Dür, Depth-first simplicial partition for copositivity detection, with an application to maxclique, Optimization Methods and Software, in press.