Signomial and polynomial optimization via relative entropy and partial dualization

Abstract

We describe a generalization of the Sums-of-AM/GM-Exponential (SAGE) methodology for relative entropy relaxations of constrained signomial and polynomial optimization problems. Our approach leverages the fact that SAGE certificates conveniently and transparently blend with convex duality, in a way which enables partial dualization of certain structured constraints. This more general approach retains key properties of ordinary SAGE relaxations (e.g. sparsity preservation), and inspires a projective method of solution recovery which respects partial dualization. We illustrate the utility of our methodology with a range of examples from the global optimization literature, along with a publicly available software package.

Appendix

As in the signomial case, Algorithm 3 always returns a vector ${\varvec{x}} \in X$. Assuming that ${\varvec{z}}$ from Line 7 are already computed as part of representing ${\varvec{{\hat{v}}}}$, the complexity of this algorithm is dominated by Line 12. The runtime of Line 12 is in turn negligible relative to solving a SAGE relaxation to obtain vectors ${\varvec{v}}$ and ${\varvec{{\hat{v}}}}$. Infeasibility errors encountered in Line 12 should be handled by jumping to Line 15.

Let us describe the ways in which Algorithm 4 differs from the discussion in Sect. 4.2.2. First- there are changes to the sets U and W. The set U now drops any rows ${\varvec{\alpha }}_i$ from ${{\varvec{\alpha }}}$ where ${\varvec{\alpha }}_i$ is even; it is easy to verify that this does not affect the set of solutions to the appropriate linear system. The set W changes by only considering j where at least one $\alpha _{ij} \equiv 1 \mod 2$. This change is valid because if $\alpha _{ij}$ is even for all i, then the sign of variable $x_j$ is irrelevant to the underlying optimization problem, and we make take $x_j \ge 0$ without loss of generality.

Next we speak to the “hueristic” sign recovery. We partly mean to leave this as open-ended, however for completeness we describe the algorithm used in sageopt. The goal is to find a vector ${\varvec{s}}$ in $\{+1,-1\}$ so that the signs of ${\varvec{s}}^{{{\varvec{\alpha }}}} \doteq ({\varvec{s}}^{{\varvec{\alpha }}_1},\ldots ,{\varvec{s}}^{{\varvec{\alpha }}_m})$ match the signs of ${\varvec{v}}$ to the greatest extent possible. However, we consider how having ${\varvec{s}}^{{\varvec{\alpha }}_i}$ match the sign of $v_i$ may not be very important if $v_i$ is very small. Therefore we use a merit function $M({\varvec{s}}) = {\varvec{v}}^\intercal {\varvec{s}}^{{{\varvec{\alpha }}}}$ to evaluate the quality of candidate signs ${\varvec{s}}$. We apply a greedy algorithm to maximize the merit function $M({\varvec{s}})$ as follows: initialize ${\varvec{s}} = {\varvec{1}}$, and a set of undecided coordinates $C = \{1,\ldots ,n\}$. As long as the set C is nonempty, find an index $i^\star \in C$ so that changing $s_{i^\star } = 1$ to $s_{i^\star } = -1$ maximizes improvement in the merit function. If the improvement is positive, then perform the update $s_{i^\star } \leftarrow -1$. Regardless of whether or not the improvement is positive, remove $i^\star$ from C. Once C is empty, return ${\varvec{s}}$.