Refined TSSOS

TL;DR

A new approach is suggested that refines TSSOS iterations using combinatorial optimization and results in block-diagonal matrices with reduced maximum block sizes, showing the large potential for computational speedup for unconstrained and constrained polynomial optimization problems.

Abstract

The moment-sum of squares hierarchy by Lasserre has become an established technique for solving polynomial optimization problems. It provides a monotonically increasing series of tight bounds, but has well-known scalability limitations. For structured optimization problems, the term-sparsity SOS (TSSOS) approach scales much better due to block-diagonal matrices, obtained by completing the connected components of adjacency graphs. This block structure can be exploited by semidefinite programming solvers, for which the overall runtime then depends heavily on the size of the largest block. However, already the first step of the TSSOS hierarchy may result in large diagonal blocks. We suggest a new approach that refines TSSOS iterations using combinatorial optimization and results in block-diagonal matrices with reduced maximum block sizes. Numerical results on a benchmark library show the large potential for computational speedup for unconstrained and constrained polynomial optimization problems, while obtaining almost identical bounds in comparison to established methods.

Figures (7)

• Figure 1: The graphs $G$, $\overline{G}$ and the matrix $\overline{B}$ from Example \ref{['ex:block_closure']}
• Figure 1: Adjacency graphs of $C_{\mathscr{A}}^{(1)}$ and $C^{(1)}$ from \ref{['ex:ex1']}.
• Figure 1: Numerical results for the chordal ($c1$ and $c2$), block ($t1$) and refined TSSOS with different parameter values $\tau \in (0, 1)$ in the unconstrained case: plots in column 1 show the fraction of problems solved to a certain accuracy by method $M$, i.e., with $\frac{|\Theta_{M} - \Theta_B|}{|\Theta_B|} \leq tol$, the ratio of CPU time for method $M$ to CPU time of TSSOS ($k=1$), i.e., $T_{M} / T_{t1}$, averaged over all problems in the set is depicted in column 2, box plots showing the spread of values $T_{M} / T_{t1}$ are given in column 3, where the central mark indicates the median, and the bottom and top edges of the box indicate the 25th and 75th percentiles, respectively. Note that the CPU time limit for the SDP solver is 5000 seconds. If for some problem the solver fails to terminate within this time on a relaxation produced by method $M$, this problem is considered to be unsolved by this method.
• Figure 1: Numerical results on polynomials from set I for the chordal ($c1$ and $c2$), block ($t1$) and refined TSSOS with different parameter values $\tau_0 = \tau_1 = \tau$, $\tau \in \set{0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7}$ with $\mathbf{K} := \{(x_1, \ldots, x_n) \in \mathbb{R}^n \; | \; g_1 = 25 - (x_1^2 + \cdots + x_n^2) \geq 0\}$ the relaxation order $\hat{d} = 4$: plots in column 1 show the fraction of 100 problems solved to a certain accuracy by method $M$, i.e., with $\frac{|\Theta_{M} - \Theta_B|}{|\Theta_B|} \leq tol$, the ratio of CPU time for method $M$ to CPU time of TSSOS ($k=1$), i.e., $T_{M} / T_{t1}$, averaged over 100 problems is depicted in column 2, box plots showing the spread of values $T_{M} / T_{t1}$ are given in column 3, where the central mark indicates the median, and the bottom and top edges of the box indicate the 25th and 75th percentiles, respectively. Note that the CPU time limit for the SDP solver is 5000 seconds. If for some problem the solver fails to terminate within this time on a relaxation produced by method $M$, this problem is considered to be unsolved by this method.
• Figure 2: Adjacency graphs of $C_{\mathscr{A}}^{(1)}$ and $C^{(1)}$ from \ref{['ex:ex2']}.

Theorems & Definitions (18)

• Example 2.1
• Remark 2.2
• Lemma 3.1
• Proof 1
• Remark 3.2
• Lemma 3.3
• Proof 2
• Example 3.4
• Example 3.5
• Proposition 4.1