Hybrid Methods in Polynomial Optimisation

TL;DR

This paper introduces a hybrid approach combining first- and second-order methods to efficiently solve large-scale polynomial optimization problems, overcoming limitations of traditional SDP solvers.

Contribution

It proposes a novel algorithm that switches between first- and second-order methods based on a quantitative criterion, enabling quadratic convergence in large-scale POPs.

Findings

Effective in solving large-scale optimal power flow problems

Achieves quadratic convergence with the proposed switching criterion

Overcomes limitations of interior-point methods for large SDPs

Abstract

The Moment/Sum-of-squares hierarchy provides a way to compute the global minimizers of polynomial optimization problems (POP), at the cost of solving a sequence of increasingly large semidefinite programs (SDPs). We consider large-scale POPs, for which interior-point methods are no longer able to solve the resulting SDPs. We propose an algorithm that combines a first-order method for solving the SDP relaxation, and a second-order method on a non-convex problem obtained from the POP. The switch from the first to the second-order method is based on a quantitative criterion, whose satisfaction ensures that Newton's method converges quadratically from its first iteration. This criterion leverages the point-estimation theory of Smale and the active-set identification. We illustrate the methodology to obtain global minimizers of large-scale optimal power flow problems.