Abstract
We consider the problem of finding the (unconstrained) global minimum of a real valued polynomial \(p(x):R^n\longrightarrow R\) . We study the problem of finding the bounds of global minimizers. It is shown that the unconstrained optimization reduces to some constrained optimizations which can be approximated by solving some convex linear matrix inequality (LMI) problems.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Lasserre J.B. (2001). Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11(3): 796–817
Shor N.Z. (1987). Quadratic optimization problems. Soviet J. Comput. Syst. Sci. 25: 1–11
Shor N.Z. (1998). Non-differentiable optimization and polynomial problems. Kluwer, Dordrecht
Ferrier C. (1998). Hilbert’s 17th problem and best dual bounds in quadratic minimization. Cybernet. Syst. Anal. 34: 696–709
Nesterov, Y.: Squared functional systems and optimization problems. In: Frenk, H., Roos, K., Terlaky, T., Zhang, S. (eds.) High Performance Optimization. Kluwer, Dordrecht (2000)
Berg, C.: The multidimensional moment problem and semi-groups. In: Landau, H.J. (ed.) Moments in Mathematics, pp. 110–124. AMS, Providence (1980)
Curto R.E. and Fialkow L.A. (1991). Recursiveness, positivity and truncated moment problems. Houston J. Math. 17: 603–635
Curto R.E. (1998). Flat extensions of positive moment matrices: Recursively generated relations. Mem. Am. Math. Soc. 136: 648
Jacobi T. (2001). A representation theorem for certain partially ordered commutative rings. Math. Z. 237(2): 259–273
Putinar M. (1993). Positive polynomial on compact semi-algebraic sets. Ind. Univ. Math. J. 42: 969–984
Putinar M. and Vasilescu F.-H. (1999). Solving moment problems by dimensional extension. Ann. Math. 149: 1087–1107
Simon B. (1998). The classical moment problem as a self-adjoint finite difference operator. Adv. Math. 137: 82–203
Schmudgen K. (1991). The K-moment problem for compact semi-algebraic sets. Math. Ann. 289: 203–206
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was partly supported by the National Science Foundation of China under grant No. 10671145.
Rights and permissions
About this article
Cite this article
Zhu, J., Zhang, X. On global optimizations with polynomials. Optimization Letters 2, 239–249 (2008). https://doi.org/10.1007/s11590-007-0054-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11590-007-0054-5