Hybrid Methods in Polynomial Optimisation
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.
Cite
@article{arxiv.2305.16122,
title = {Hybrid Methods in Polynomial Optimisation},
author = {Johannes Aspman and Gilles Bareilles and Vyacheslav Kungurtsev and Jakub Marecek and Martin Takáč},
journal= {arXiv preprint arXiv:2305.16122},
year = {2023}
}