Semialgebraic set

Abstract: We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equations and inequalities, which is NP-hard in general. Hierarchies of semidefinite relaxations have been proposed in the literature, involving positive semidefinite moment matrices and the dual theory of sums of squares of polynomials. We present these hierarchies of approximations and their main properties: asymptotic/finite convergence, optimality certificate, and extraction of global optimum solutions. We review the mathematical tools underlying these properties, in particular, some sums of squares representation results for positive polynomials, some results about moment matrices (in particular, of Curto and Fialkow), and the algebraic eigenvalue method for solving zero-dimensional systems of polynomial equations. We try whenever possible to provide detailed proofs and background.

Abstract: In this chapter we begin the study of the multidimensional moment problem. The passage to dimensions d ≥ 2 brings new difficulties and unexpected phenomena. In Sect. 3.2 we derived solvability criteria of the moment problem on intervals in terms of positivity conditions. It seems to be natural to look for similar characterizations in higher dimensions as well. This leads us immediately into the realm of real algebraic geometry and to descriptions of positive polynomials on semi-algebraic sets. In this chapter we treat this approach for basic closed compact semi-algebraic subsets of \(\mathbb{R}^{d}\). It turns out that for such sets there is a close interaction between the moment problem and Positivstellensatze for strictly positive polynomials.

Abstract: We consider a polynomial programming problem $\P$ on a compact basic semialgebraic set $\K\subset\R^n$, described by $m$ polynomial inequalities $g_j(X)\geq0$, and with criterion $f\in\R[X]$. We propose a hierarchy of semidefinite relaxations in the spirit of those of Waki e [SIAM J. Optim., 17 (2006), pp. 218-242]. In particular, the SDP-relaxation of order $r$ has the following two features: (a) The number of variables is $O(\kappa^{2r})$, where $\kappa=\max[\kappa_1,\kappa_2]$ with $\kappa_1$ (resp., $\kappa_2$) being the maximum number of variables appearing in the monomials of $f$ (resp., appearing in a single constraint $g_j(X)\geq0$). (b) The largest size of the linear matrix inequalities (LMIs) is $O(\kappa^r)$. This is to compare with the respective number of variables $O(n^{2r})$ and LMI size $O(n^r)$ in the original SDP-relaxations defined in [J. B. Lasserre, SIAM J. Optim., 11 (2001), pp. 796-817]. Therefore, great computational savings are expected in case of sparsity in the data $\{g_j,f\}$, i.e., when k is small, a frequent case in practical applications of interest. The novelty with respect to [H. Waki, S. Kim, M. Kojima, and M. Maramatsu, SIAM J. Optim., 17 (2006), pp. 218-242] is that we prove convergence to the global optimum of $\P$ when the sparsity pattern satisfies a condition often encountered in large size problems of practical applications, and known as the running intersection property in graph theory. In such cases, and as a by-product, we also obtain a new representation result for polynomials positive on a compact basic semialgebraic set, a sparse version of Putinar’s Positivstellensatz [M. Putinar, Indiana Univ. Math. J., 42 (1993), pp. 969-984].