Abstract
A standard quadratic optimization problem consists in minimizing a quadratic form over the standard simplex. This problem has a closer connection with copositive optimization, where the copositivity appears in local optimality conditions. In this paper, we establish a relationship between the sparsity of a solution of the standard quadratic optimization problem, from one hand, and the copositivity of the Hessian matrix of the Lagrangian from other hand. More precisely, if the Hessian matrix is copositive we prove that the number of zero components, of a local minimizer associated with a standard quadratic optimization problem, is greater or equal to the number of negative eigenvalues counting multiplicities. Moreover, we show that if the number of zero components of a local minimizer is equal to the number of negative eigenvalues of the Hessian matrix, then the strict complementarity condition is satisfied and the critical cone is a linear subspace.
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References
Motzkin, T.S.: Copositive quadratic forms. Natl. Bureau Stand. Rep. 1818, 11–22 (1952)
Burer, S.: On the copositive representation of binary and continuous nonconvex quadratic programs. Math. Program. 120(2), 479–495 (2009)
Jacobson, D.H.: Extensions of Linear-Quadratic Control, Optimization and Matrix Theory. Academic, London (1977)
Jones, P.C.: A note on the Talman, Van der Heyden linear complementarity algorithm. Math. Program. 25, 122–124 (1983)
Johnson, C.R., Reams, R.: Spectral theory of copositive matrices. Linear Algebra Appl. 395, 275–281 (2005)
Kaplan, W.: A test for copositive matrices. Linear Algebra Appl. 313(1–3), 203–206 (2000)
Jargalsaikhan, B.: Indefinite copositive matrices with exactly one positive eigenvalue or exactly one negative eigenvalue. Electron. J. Linear Algebra. 26, 754–761 (2013)
Li, P., Feng, Y.Y.: Criteria for copositive matrices of order four. Linear Algebra Appl. 194, 109–124 (1993)
Bundfuss, S., Dür, M.: Algorithmic copositivity detection by simplicial partition. Linear Algebra Appl. 428, 1511–1523 (2008)
Dür, M.: Copositive programming—a survey. In: Diehl, M., Glineur, F., Jarlebring, E., Michiels, W. (eds.) Recent Advances in Optimization and its Applications in Engineering, pp. 3–20. Springer, Berlin (2010)
Bomze, I.M., Eichfelder, G.: Copositivity detection by difference-of-convex decomposition and \(\omega \)-subdivision. Math. Program. 138(1–2), 365–400 (2013)
Väliaho, H.: Quadratic-programming criteria for copositive matrices. Linear Algebra Appl. 119, 163–182 (1989)
Väliaho, H.: Criteria for copositive matrices. Linear Algebra Appl. 81, 19–34 (1986)
Bomze, I.M.: Copositive optimization-recent developments and applications. Eur. J. Oper. Res. 216(3), 509–520 (2012)
Bomze, I.M., Schachinger, W., Uchida, G.: Think co(mpletely)positive ! Matrix properties, examples and a clustered bibliography on copositive optimization. J. Global Optim. 52(3), 423–445 (2012)
Hiriart-Urruty, J.B., Seeger, A.: A variational approach to copositive matrices. SIAM Rev. 52(4), 593–629 (2010)
Micchelli, C.A., Pinkus, A.: Some remarks on nonnegative polynomials on polyhedra. In: Anderson, T.W., Athreya, K.B., Iglehart, D.L. (eds.) Probability, Statistics, and Mathematics Papers in Honor of Samuel Karlin, pp. 163–186. Academic, Boston (1989)
Borwein, J.M.: Necessary and sufficient conditions for quadratic minimality. Numer. Funct. Anal. Optim. 5(2), 127–140 (1982)
Hoffman, A.J., Pereira, F.: On copositive matrices with \(-1\), 0, 1 entries. J. Comb. Theory Ser. A. 14(3), 302–309 (1973)
Chabrillac, Y., Crouzeix, J.P.: Definiteness and semidefiniteness of quadratic forms revisited. Linear Algebra Appl. 63, 283–292 (1984)
Han, S.P., Fujiwara, O.: An inertia theorem for symmetric matrices and its applications to nonlinear programming. Linear Algebra Appl. 72, 47–58 (1985)
Jongen, H.T., Möbert, T., Rückmann, J., Tammer, K.: On inertia and Schur complement in optimization. Linear Algebra Appl. 95, 97–109 (1987)
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Baccari, A., Naffouti, M. Copositivity and Sparsity Relations Using Spectral Properties. J Optim Theory Appl 171, 998–1007 (2016). https://doi.org/10.1007/s10957-016-0997-8
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DOI: https://doi.org/10.1007/s10957-016-0997-8