Abstract
In this paper we present necessary conditions for global optimality for polynomial problems with box or bivalent constraints using separable polynomial relaxations. We achieve this by first deriving a numerically checkable characterization of global optimality for separable polynomial problems with box as well as bivalent constraints. Our necessary optimality conditions can be numerically checked by solving semi-definite programming problems. Then, by employing separable polynomial under-estimators, we establish sufficient conditions for global optimality for classes of polynomial optimization problems with box or bivalent constraints. We construct underestimators using the sum of squares convex (SOS-convex) polynomials of real algebraic geometry. An important feature of SOS-convexity that is generally not shared by the standard convexity is that whether a polynomial is SOS-convex or not can be checked by solving a semidefinite programming problem. We illustrate the versatility of our optimality conditions by simple numerical examples.
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References
Ahmadi, A.A., Parrilo, P.: A convex polynomial that is not SOS-convex. Math. Program. 135, 275–292 (2012)
Anjos, M., Lasserre, J.B.: Handbook of Semidefinite, Conic and Polynomial Optimization. Springer, Berlin (2012)
Beck, A., Teboulle, M.: Global optimality conditions for quadratic optimization problems with binary constraints. SIAM. J. Optim. 11, 179–188 (2000)
Bertsimas, D., Popescu, I.: Optimal inequalities in probability theory: a convex optimization approach. SIAM J. Optim. 15(3), 780–804 (2005)
Blekherman, G., Parrilo, P.A., Thomas, R.: Semidefinite Optimization and Convex Algebraic Geometry, SIAM Publications, (2012)
Helton, J.W., Nie, J.W.: Semidefinite representation of convex sets. Math. Program, 122(1), Ser. A, 21–64 (2010)
Hiriart-Urruty, J.B.: Global optimality conditions in maximizing a convex quadratic function under convex quadratic constraints. J. Global Optim. 21, 445–455 (2001)
Horst, R., Pardalos, M.: Handbook of global optimization. In: Horst, R., Pardalos, P.M. (eds.) Nonconvex Optimization and its Applications, 3 Kluwer Academic Publishers, Dordrechet, 1995
Jeyakumar, V., Rubinov, A.M., Wu, Z.Y.: Non-convex quadratic minimization problems with quadratic constraints: global optimality conditions. Math. Program. Ser. A 110, 521–541 (2007)
Jeyakumar, V., Srisatkunarajah, S., Huy, N.Q.: Kuhn–Tucker sufficiency for global minimum of multi-extremal mathematical programming problems. J. Math. Anal. Appl. 335, 779–788 (2007)
Jeyakumar, V., Lee, G.M., Li, G.Y.: Alternative theorems for quadratic inequality systems and global quadratic optimization. SIAM J. Optim. 20(2), 983–1001 (2009)
Jeyakumar, V., Huy, N.Q., Li, G.: Necessary and sufficient conditions for S-lemma and nonconvex quadratic optimization. Optim. Eng. 10(4), 491–503 (2009)
Jeyakumar, V., Li, G.Y.: Necessary global optimality conditions for nonlinear programming problems with polynomial constraints. Math. Program. 126, 393–399 (2011)
Jeyakumar, V., Li, G.: Regularized Lagrangian duality for linearly constrained quadratic optimization and trust-region problems. J. Global Optim. 49(1), 1–14 (2011)
Lasserre, J.B.: Polynomials nonnegative on a grid and discrete optimization. Trans. Am. Math. Soc. 354(2), 631–649 (2002)
Lasserre, J.B.: Moments, Positive Polynomials and Their Applications. Imperial College Press (2009)
Laurent, M.: Sums of squares, moment matrices and optimization over polynomials. In: Putinar, M., Sullivant, S. (eds.) Emerging Applications of Algebraic Geometry, vol. 149 of IMA Volumes in Mathematics and its Applications, pp. 157–270. Springer, Berlin (2009)
Li, G.: Global quadratic minimization over bivalent constraints: necessary and sufficient global optimality condition. J. Optim. Theory Appl. 152(3), 710–726 (2012)
Löfberg, J.: YALMIP : a toolbox for modeling and optimization in MATLAB. In: Proceedings of the CACSD Conference, Taipei, Taiwan (2004)
Löfberg, J.: Pre- and post-processing sum-of-squares programs in practice. IEEE Trans. Autom. Control 54, 1007–1011 (2009)
Neumaier, A.: Second-order sufficient optimality conditions for local and global nonlinear programming. J. Global Optim. 9, 141–151 (1996)
Parrilo, P.A.: Semidefinite programming relaxations for semialgebraic problems. Math. Program. Ser. B 96, 293–320 (2003)
Phillips, A.T., Rosen, J.B.: A quadratic assignment formulation of the molecular confirmation problem. J. Global Optim. 4, 229–241 (1994)
Pinar, M.C.: Sufficient global optimality conditions for bivalent quadratic optimization. J. Optim. Theory Appl. 122, 433–440 (2004)
Quan, J., Wu, Z.Y., Li, G.Q.: Global optimality conditions for some classes of polynomial integer programming problems. J. Ind. Manag. Optim. 7, 67–78 (2011)
Wang, Y., Liang, Z.: Global optimality conditions for cubic minimization problem with box or binary constraits. J. Global Optim 47, 583–595 (2010)
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The work of this author was completed while he was visiting the University of New South Wales, Sydney, Australia.
The authors are grateful to the referees and the editor for their constructive comments and helpful suggestions which have contributed to the final preparation of the paper. Research was partially supported by a grant from the Australian Research Council.
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Jeyakumar, V., Li, G. & Srisatkunarajah, S. Global optimality principles for polynomial optimization over box or bivalent constraints by separable polynomial approximations. J Glob Optim 58, 31–50 (2014). https://doi.org/10.1007/s10898-013-0058-x
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DOI: https://doi.org/10.1007/s10898-013-0058-x