A Fine-Grained Variant of the Hierarchy of Lasserre

TL;DR

This paper introduces a more detailed variant of the Lasserre hierarchy for polynomial optimization, along with efficient first-order methods that enable better warm-starting from lower levels, improving practical solvability.

Contribution

A new fine-grained variant of the Lasserre hierarchy is proposed, coupled with first-order algorithms for efficient solution warm-starting from lower hierarchy levels.

Findings

Enhanced convergence properties demonstrated

Efficient warm-starting reduces computational effort

Applicable to larger polynomial optimization problems

Abstract

There has been much recent interest in hierarchies of progressively stronger convexifications of polynomial optimisation problems (POP). These often converge to the global optimum of the POP, asymptotically, but prove challenging to solve beyond the first level in the hierarchy for modest instances. We present a finer-grained variant of the Lasserre hierarchy, together with first-order methods for solving the convexifications, which allow for efficient warm-starting with solutions from lower levels in the hierarchy.