Global optimization of polynomials restricted to a smooth variety using sums of squares

  • Authors:

  • Aurélien Greuet;Feng Guo;Mohab Safey El Din;Lihong Zhi

  • Affiliations:

  • UPMC, Université Paris 06, INRIA, Paris Rocquencourt Center, SALSA Project, LIP6/CNRS, UMR 7606, France and Laboratoire de Mathématiques (LMV-UMR8100), Université de Versailles-Sain ...;Key Laboratory of Mathematics Mechanization, AMSS, Beijing 100190, China;UPMC, Université Paris 06, INRIA, Paris Rocquencourt Center, SALSA Project, LIP6/CNRS, UMR 7606, France;Key Laboratory of Mathematics Mechanization, AMSS, Beijing 100190, China

  • Venue:
  • Journal of Symbolic Computation
  • Year:
  • 2012

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Abstract

Let f"1,...,f"p be in Q[X], where X=(X"1,...,X"n)^t, that generate a radical ideal and let V be their complex zero-set. Assume that V is smooth and equidimensional. Given f@?Q[X] bounded below, consider the optimization problem of computing f^@?=inf"x"@?"V"@?"R"^"nf(x). For A@?GL"n(C), we denote by f^A the polynomial f(AX) and by V^A the complex zero-set of f"1^A,...,f"p^A. We construct families of polynomials M"0^A,...,M"d^A in Q[X]: each M"i^A is related to the section of a linear subspace with the critical locus of a linear projection. We prove that there exists a non-empty Zariski-open set @?@?GL"n(C) such that for all A@?@?@?GL"n(Q), f(x) is positive for all x@?V@?R^n if and only if, f^A can be expressed as a sum of squares of polynomials on the truncated variety generated by the ideal , for 0@?i@?d. Hence, we can obtain algebraic certificates for lower bounds on f^@? using semi-definite programs. Some numerical experiments are given. We also discuss how to decrease the number of polynomials in M"i^A.