Representations of positive polynomials and optimization on noncompact semialgebraic sets

摘要

This paper studies the representation of a positive polynomial f on a closed semialgebraic set S: = {κ ∈ R~n | gi(κ) = 0, i = 1, . . ., l, hj(κ) ≥ 0, j = 1, . . .,m} modulo the so-called critical ideal I(f, S) of f on S. Under a constraint qualification condition, it is demonstrated that, if either f > 0 on S or f ? 0 on S and the critical ideal I(f, S) is radical, then f belongs to the preordering generated by the polynomials h1, . . ., hm modulo the critical ideal I(f, S). These facts imply that we can find a natural sequence of semidefinite programs whose optimal values converge monotonically, increasing to the infimum value f?: = inf κ∈S f(κ) of f on S, provided that the infimum value is attained at some point. Besides, we shall construct a finite set in R containing the infimum value f?. Moreover, some relations between the Fedoryuk [Soviet Math. Dokl., 17 (1976), pp. 486-490] and Malgrange [Complex Analysis, Microlocal Calculus and Relativistic Quantum Theory, Lecture Notes in Phys. 126, Springer, Berlin, 1980, pp. 170-177] conditions and coercivity for polynomials, which are bounded from below on S, are also established. In particular, a sufficient condition for f to attain its infimum on S is derived from these facts. We also show that every polynomial f, which is bounded from below on S, can be approximated in the l1-norm of coefficients by a sequence of polynomials f∈ that are coercive. Finally, it is shown that almost every linear polynomial function, which is bounded from below on S, attains its infimum on S and has the same asymptotic growth at infinity.