Hybrid Methods in Polynomial Optimisation

Johannes Aspman; Gilles Bareilles; Vyacheslav Kungurtsev; Jakub Marecek; Martin Takáč

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arXiv · 2305.16122

Hybrid Methods in Polynomial Optimisation

Johannes Aspman·Gilles Bareilles·Vyacheslav Kungurtsev·Jakub Marecek·Martin Takáč

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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.

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Johannes Aspman, Gilles Bareilles, Vyacheslav Kungurtsev, Jakub Marecek, Martin Takáč. 2023-05-25. Hybrid Methods in Polynomial Optimisation. https://arxiv.org/abs/2305.16122

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