# A scalable algorithm for sparse and robust portfolios

### Abstract

A central problem in financial economics concerns investing capital to maximize a portfolio's expected return while minimizing its variance, subject to an upper bound on the number of positions, minimum investment and linear inequality constraints, at scale. Existing approaches to this problem do not provide provably optimal portfolios for real-world problem sizes with more than 300 securities. In this paper, we present a cutting-plane method which solves problems with $1000$s of securities to provable optimality, by exploiting a dual representation of the continuous Markowitz problem to obtain a closed form representation of the problem's subgradients. We improve the performance of the cutting-plane method in three different ways. First, we implement a local-search heuristic which applies our subgradient representation to obtain high-quality warm-starts. Second, we embed the local-search heuristic within the cutting-plane method to accelerate recovery of an optimal solution. Third, we exploit a correspondence between the convexified sparse Markowitz problem and a rotated second-order cone problem to obtain a tight lower bound which is sometimes exact. Finally, we establish that the cutting-plane method is 3-4 orders of magnitude more efficient than existing methods, and construct provably optimal sparsity-constrained frontiers for the S&P 500, Russell 1000, and Wilshire 5000.

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