Representations of Positive Polynomials and Optimization on Noncompact Semialgebraic Sets

• Authors: Hà Huy Vui;Pham Tien So'n
• Venue: SIAM Journal on Optimization
• Year: 2010

This paper studies the representation of a positive polynomial $f$ on a closed semialgebraic set $S:=\{x\in\mathbb{R}^n\mid g_i(x)=0, i=1,\dots,l, h_j(x)\geq0, j=1,\dots,m\}$ modulo the so-called critical ideal $I(f,S)$ of $f$ on $S$. Under a constraint qualification condition, it is demonstrated that, if either $f0$ on $S$ or $f\geq0$ on $S$ and the critical ideal $I(f,S)$ is radical, then $f$ belongs to the preordering generated by the polynomials $h_1,\dots,h_m$ 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 $f^*:=\inf_{x\in S}f(x)$ of $f$ on $S$, provided that the infimum value is attained at some point. Besides, we shall construct a finite set in $\mathbb{R}$ 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 $l_1$-norm of coefficients by a sequence of polynomials $f_\epsilon$ 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.