Semidefinite programming relaxation methods for global optimization problems with sparse polynomials and unbounded semialgebraic feasible sets

Abstract

We propose a hierarchy of semidefinite programming (SDP) relaxations for polynomial optimization with sparse patterns over unbounded feasible sets. The convergence of the proposed SDP hierarchy is established for a class of polynomial optimization problems. This is done by employing known sums-of-squares sparsity techniques of Kojima and Muramatsu Comput Optim Appl 42(1):31–41, (2009) and Lasserre SIAM J Optim 17:822–843, (2006) together with a representation theorem for polynomials over unbounded sets obtained recently in Jeyakumar et al. J Optim Theory Appl 163(3):707–718, (2014). We demonstrate that the proposed sparse SDP hierarchy can solve some classes of large scale polynomial optimization problems with unbounded feasible sets using the polynomial optimization solver SparsePOP developed by Waki et al. ACM Trans Math Softw 35:15 (2008).

Appendix

The polynomial optimization problem $(P_0)$ has a close relationship with the problem of finding the solution with the least number of nonzero components which satisfies a system of polynomial inequalities and simple bounds. Mathematically, the problem of finding the solution with the least number of nonzero components which satisfies a system of polynomial inequalities and simple bounds, can be formulated as

where $n_l \in \mathbb {N}$, $l=1,\ldots ,q$, and $\Vert x\Vert _0$ denotes the $l_0$-seminorm of the vector $x \in \mathbb {R}^n$, which gives the number of nonzero components of the vector x.

In the case where $q=1$, $I_1=\{1,\ldots ,n\}$, $g_j(x)=a_j^Tx-b_j$, $j=1,\ldots ,m$, $g_{j}(x)=-(a_j^Tx-b_j)$, $j=m+1,\ldots ,2m$, the problem $(P_0')$ collapses to the sparse optimization problem which finds the solution with the least number of nonzero components satisfying simple bounds as well as linear equations $Ax=b$ with more unknowns than equalities:

where $A=(a_1,\ldots ,a_m)^T \in \mathbb {R}^{m \times n}$ $(m \le n)$, $b=(b_1,\ldots ,b_m)^T \in \mathbb {R}^m$. We note that the standard sparse optimization problem which is given by

arises in signal processing and was examined, for example, [2–4, 33]. Moreover, problems $(P_1)$ and $(P_2)$ have the same optimal value if $M> \Vert x^*\Vert _2^2$ for some solution $x^*$ of problem $(P_2)$.

In fact, the problem $(P_0)$ and problem $(P_0')$ are equivalent in the sense that $\min (P_0)=\min (P_0')$ and $(x^{1*}, \ldots , x^{q*})\in \mathbb {R}^{n_1} \times \cdots \times \mathbb {R}^{n_q}$ is a solution of problem $(P_0)$ if and only if $(x^{l*},w^{l*}) \in \mathbb {R}^{n_l} \times \mathbb {R}^{n_l}$, $l=1,\ldots ,q$, is a solution of problem $(P_0')$ where $w^{l*}=(w^{l*}_{i_1},\ldots ,w^{l*}_{i_{n_l}}) \in \mathbb {R}^{n_l}$ is defined by

$$\begin{aligned} w^{l*}_i=\left\{ \begin{array}{lll} 1 &{} \quad \text{ if } &{} x_i^{l*} \ne 0, \\ 0 &{} \quad \text{ if } &{} x_i^{l*} = 0. \end{array}\right. \ \ \ \ \ \ i \in I_l, \ l=1,\ldots ,q. \end{aligned}$$

(4.3)