Abstract
In this work we propose a general procedure for analyzing global minima of arbitrary mathematical programs which is based in probability measures and moments theory. We give a general characterization of global minima of arbitrary programs, and as a particular case, we characterize the global minima of unconstrained one dimensional polynomial programs by using a particular semidefinite program.
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References
Ben-Tal, A. and A. Nemirovski, Lectures on Modern Convex Optimization, MPS-SIAM, 2001.
Berg, C. et al., A remark on the multidimensional moment problem, Math. Ann. 223, p.163–169, 1979.
Boyd, S. et al., Linear Matrix Inequalities and Control Theory, SIAM, 1994.
Curto, R. and L.A. Fialkow, Recursiveness, positivity and truncated moment problems, Houston Journal of Mathematics, vol. 17, No. 4, 1991.
Gahinet, P. et al., LMI Control Toolbox User’s Guide, The MathWorks Inc., 1995.
Krein, M.G. and A.A. Nudel’man, The Markov Moment Problem and Extremal Problems, Translations of Mathematical Monographs, vol. 50, AMS, 1977.
Lasserre, J., Semidefinite programming vs 1p relaxations for polynomial programming, Mathematics of Operations Research Journal, vol. 27, No 2, pp. 347–360, 2002.
Lasserre, J., Global optimization with polynomials and the problem of moments, SIAM J. Optim., vol 11, No 3, pp. 796–817, 2001.
Lasserre, J., New positive semidefinite relaxations for nonconvex quadratic programs, in Advances in Convex Analysis and Global Optimization, Non Convex Optimization and Its Applications Series, vol. 54, Kluwer, 2001.
Meziat, R., The method of moments in global optimization, Journal of Mathematical Sciences, vol 116, No3, pp. 3303–3324, Kluwer, 2003.
Meziat, R., P. Pedregal and J.J. Egozcue, From a nonlinear, nonconvex variational problem to a linear, convex formulation, J. Appl. Math. Optm., vol 47, pp. 27–44, Springer Verlag, New York, 2003.
Meziat, R., P. Pedregal and J.J. Egozcue, The method of moments for non convex variational problems, in Advances in Convex Analysis and Global Optimization, Non Convex Optimization and Its Applications Series, vol. 54, pp. 371–382, Kluwer, 2001.
Meziat, R., Two dimensional non convex variational problems, to appear in the proceedings of the International Workshop in Control and Optimization, Erice, Italy, 2001.
Nesterov Y., Squared functional systems and optimization problems, in: High Performance Optimization, H. Frenk, K. Roos, T. Terlaky, S. Zhang eds, Kluwer Academic Publishers, Dordrecht, 2000.
Putinar, M., Positive polynomials on compact semi-algebraic sets, Indiana University Mathematics Journal, vol. 42, No. 3, pp. 969–984, 1993.
Shoat, J.A. and J.D. Tamarkin, The Problem of Moments, Mathematical Surveys 1, AMS, 1943.
Shor, N.Z., Nondifferentiable Optimization and Polynomial Problems, Kluwer, 1998.
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Meziat, R.J. (2004). Analysis of Non Convex Polynomial Programs by the Method of Moments. In: Floudas, C.A., Pardalos, P. (eds) Frontiers in Global Optimization. Nonconvex Optimization and Its Applications, vol 74. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0251-3_19
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DOI: https://doi.org/10.1007/978-1-4613-0251-3_19
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