Memory-efficient structured convex optimization via extreme point sampling

Nimita Shinde; Vishnu Narayanan; James Saunderson

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arXiv · 2006.10945

Memory-efficient structured convex optimization via extreme point sampling

Nimita Shinde·Vishnu Narayanan·James Saunderson

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Abstract

Memory is a key computational bottleneck when solving large-scale convex optimization problems such as semidefinite programs (SDPs). In this paper, we focus on the regime in which storing an $n\times n$ matrix decision variable is prohibitive. To solve SDPs in this regime, we develop a randomized algorithm that returns a random vector whose covariance matrix is near-feasible and near-optimal for the SDP. We show how to develop such an algorithm by modifying the Frank-Wolfe algorithm to systematically replace the matrix iterates with random vectors. As an application of this approach, we show how to implement the Goemans-Williamson approximation algorithm for \textsc{MaxCut} using $\mathcal{O}(n)$ memory in addition to the memory required to store the problem instance. We then extend our approach to deal with a broader range of structured convex optimization problems, replacing decision variables with random extreme points of the feasible region.

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Nimita Shinde, Vishnu Narayanan, James Saunderson. 2020-06-19. Memory-efficient structured convex optimization via extreme point sampling. https://arxiv.org/abs/2006.10945

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