Convergent SDP‐Relaxations in Polynomial Optimization with Sparsity 论文

摘要

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 et al. [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 κ 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].