Representations of Positive Polynomials and Optimization on Noncompact Semialgebraic Sets

This paper studies the representation of a positive polynomial f on a closed semialgebraic set S:=ⁿ gᵢ(x)=0, i=1,,l, hⱼ(x)≥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^*:=infx∈ Sf(x) 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_ε 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.