On equivalent representations and properties of faces of the cone of copositive matrices. (English) Zbl 1500.15029
Summary: The paper is devoted to the study of the cone \(\mathcal{C}OP^p\) of copositive matrices. Based on the concept of immobile indices known from semi-infinite optimization, we define zero and minimal zero vectors of a subset of the cone \(\mathcal{C}OP^p\) and use them to obtain different representations of the faces of \(\mathcal{C}OP^p\) and the corresponding dual cones. The minimal face of \(\mathcal{C}OP^p\) containing a given convex subset of this cone is described, and some propositions are proved that allow obtaining equivalent descriptions of the feasible sets of copositive problems.
MSC:
| 15B48 | Positive matrices and their generalizations; cones of matrices |
| 90C25 | Convex programming |
| 90C34 | Semi-infinite programming |
| 90C46 | Optimality conditions and duality in mathematical programming |
Keywords:
copositive programming; zero vectors of a subset of copositive matrices; regularization; face reduction; Slater conditionReferences:
| [1] | Ahmed, F.; Dür, M.; Still, G., Copositive programming via semi-infinite optimization, J Optim Theory Appl, 159, 322-340 (2013) · Zbl 1282.90203 |
| [2] | Bomze, IM; Schachinger, W.; Uchida, G., Think co(mpletely)positive ! matrix properties, examples and a clustered bibliography on copositive optimization, J Global Optim, 52, 423-445 (2012) · Zbl 1268.90051 |
| [3] | Bomze, IM., Copositive optimization – recent developments and applications, EJOR, 216, 3, 509-520 (2012) · Zbl 1262.90129 |
| [4] | Dür, M. Copositive programming – a survey. In: Diehl M, Glineur F, Jarlebring E, et al., editors. Recent advances in optimization and its applications in engineering. Berlin, Heidelberg: Springer-Verlag; 2010. p. 535. |
| [5] | Bomze, IM., Copositivity for second-order optimality conditions in general smooth optimization problems, Optimization, 65, 4, 779-795 (2016) · Zbl 1384.90074 |
| [6] | Burer, S. Copositive programming. In: Anjos MF, Lasserre JB, editors. Handbook of semidefinite, conic and polynomial optimization. Springer US; 2012. p. 201-218. (International Series in Operational Research and Management Science; 166). · Zbl 1235.90002 |
| [7] | Anjos MF, Lasserre JB, editors. Handbook of semidefinite, conic and polynomial optimization. Springer US; 2012. (International Series in Operational Research and Management Science; vol. 166). · Zbl 1235.90002 |
| [8] | Hettich, R.; Kortanek, KO., Semi-infinite programming: theory, methods, and applications, SIAM Rev, 35, 380-429 (1993) · Zbl 0784.90090 |
| [9] | Wolkowicz, H.; Saigal, R.; Vandenberghe, L., Handbook of semidefinite programming – theory, algorithms, and applications (2000), Boston: Kluwer Academic Publishers, Boston · Zbl 0962.90001 |
| [10] | Liu, YL; Pan, SH., Strong calmness of perturbed KKT system for a class of conic programming with degenerate solutions, Optimization, 68, 6, 1131-1156 (2019) · Zbl 1415.49020 |
| [11] | Lopez, M.; Still, G., Semi-infinite programming, Eur J Oper Res, 180, 2, 491-518 (2007) · Zbl 1124.90042 |
| [12] | Jongen, HT; Twilt, F.; Weber, G-W., Semi-infinite optimization: structure and stability of the feasible set, J Optim Theory Appl, 72, 529-552 (1992) · Zbl 0807.90113 |
| [13] | Klatte, D, Henrion, R. Regularity and stability in nonlinear semi-infinite optimization. In: Reemstsen R, Ruckmann JJ, editors. Semi-infinite programming. Dordrecht: Kluwer; 1998. p. 69-102. · Zbl 0911.90330 |
| [14] | Mordukhovich, B.; Nghia, TTA., Constraint qualifications and optimality conditions for nonconvex semi-infinite and infinite programs, Math Program Ser B, 139, 271-300 (2013) · Zbl 1292.90283 |
| [15] | Ramana, MV., An exact duality theory for semidefinite programming and its complexity implications, Math Prog, 77, 129-162 (1997) · Zbl 0890.90144 |
| [16] | Ramana, MV; Tuncel, L.; Wolkowicz, H., Strong duality for semidefinite programming, SIAM J Optim, 7, 3, 641-662 (1997) · Zbl 0891.90129 |
| [17] | Drusvyatskiy, D, Wolkowicz, H. The many faces of degeneracy in conic optimization. Now Publishers Inc.; 2017. p. 77-170. (Foundations and Trends in Optimization; vol. 3). |
| [18] | Waki, H.; Muramatsu, M., Facial reduction algorithms for conic optimization problems, J Optim Theory Appl, 158, 188-215 (2013) · Zbl 1272.90048 |
| [19] | Berman, A.; Dür, M.; Shaked-Monderer, N., Open problems in the theory of completely positive and copositive matrices, Electron J Linear Algebra, 29, 46-58 (2015) · Zbl 1326.15048 |
| [20] | Afonin, A.; Hildebrand, R.; Dickinson, P., The extreme rays of the \(####\) copositive cone, J Global Optim, 79, 153-190 (2020) · Zbl 1465.90066 |
| [21] | Hildebrand, R., The extreme rays of the \(####\) copositive cone, Linear Algebra Appl, 437, 7, 1538-1547 (2012) · Zbl 1259.15045 |
| [22] | Dickinson, PJC., Geometry of the copositive and completely positive cones, J Math Anal Appl, 380, 377-395 (2011) · Zbl 1229.90195 |
| [23] | Dickinson, PJC; Hildebrand, R., Considering copositivity locally, J Math Anal Appl, 437, 2, 1184-1195 (2016) · Zbl 1382.15048 |
| [24] | Bundfuss, S.; Dür, M., Algorithmic copositivity detection by simplicial partition, Linear Algebra Appl, 428, 1511-1523 (2008) · Zbl 1138.15007 |
| [25] | Kostyukova, OI; Tchemisova, TV., Optimality conditions for convex semi-infinite programming problems with finitely representable compact index sets, J Optim Theory Appl, 175, 1, 76-103 (2017) · Zbl 1386.90104 |
| [26] | Kostyukova, OI; Tchemisova, TV; Dudina, OS., Immobile indices and CQ-free optimality criteria for linear copositive programming problems, Set-Valued Var Anal, 28, 89-107 (2020) · Zbl 1442.90145 |
| [27] | Kostyukova, OI; Tchemisova, TV., CQ-free optimality conditions and strong dual formulations for a special conic optimization problem, SOIC, 3, 3, 668-683 (2020) |
| [28] | Pataki, G., On the connection of facially exposed and nice cones, J Math Anal Appl, 400, 1, 211-221 (2013) · Zbl 1267.90098 |
| [29] | Rockafellar, RT., Convex analysis (1970), Princeton: Princeton University Press · Zbl 0193.18401 |
| [30] | Eichfelder, G.; Povh, J., On the set-semidefinite representation of nonconvex quadratic programs over arbitrary feasible sets, Optim Lett, 7, 1373-1386 (2013) · Zbl 1298.90064 |
| [31] | Pataki, G. The geometry of semidefinite programming. In: Wolkowicz H, Saigal R, Vandenberghe L, editors. Handbook of semidefinite programming. Amsterdam: Kluwers Academic; 2000. p. 29-65. · Zbl 0957.90531 |
| [32] | Tunçel, L.; Wolkowicz, H., Strong duality and minimal representations for cone optimization, Comput Optim Appl, 53, 2, 619-648 (2012) · Zbl 1284.90080 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
