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arXiv · 1803.10928

Design of First-Order Optimization Algorithms via Sum-of-Squares Programming

Mahyar Fazlyab·Manfred Morari·Victor M. Preciado
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Abstract

In this paper, we propose a framework based on sum-of-squares programming to design iterative first-order optimization algorithms for smooth and strongly convex problems. Our starting point is to develop a polynomial matrix inequality as a sufficient condition for exponential convergence of the algorithm. The entries of this matrix are polynomial functions of the unknown parameters (exponential decay rate, stepsize, momentum coefficient, etc.). We then formulate a polynomial optimization, in which the objective is to optimize the exponential decay rate over the parameters of the algorithm. Finally, we use sum-of-squares programming as a tractable relaxation of the proposed polynomial optimization problem. We illustrate the utility of the proposed framework by designing a first-order algorithm that shares the same structure as Nesterov's accelerated gradient method.

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Mahyar Fazlyab, Manfred Morari, Victor M. Preciado. 2018-03-29. Design of First-Order Optimization Algorithms via Sum-of-Squares Programming. https://arxiv.org/abs/1803.10928

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