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Mean-reverting portfolios with few assets, but high variance, are of greatinterest for investors in financial markets. Such portfolios arestraightforwardly profitable because they include a small number of assetswhose prices not only oscillate predictably around a long-term mean but alsopossess enough volatility. Roughly speaking, sparsity minimizes trading costs,volatility provides arbitrage opportunities, and mean-reversion property equipsinvestors with ideal investment strategies. Finding such favorable portfolioscan be formulated as a nonconvex quadratic optimization problem with anadditional sparsity constraint. To the best of our knowledge, there is nomethod for solving this problem and enjoying favorable theoretical propertiesyet. In this paper, we develop an effective two-stage algorithm for thisproblem. In the first stage, we apply a tailored penalty decomposition methodfor finding a stationary point of this nonconvex problem. For a fixed penaltyparameter, the block coordinate descent method is utilized to find a stationarypoint of the associated penalty subproblem. In the second stage, we improve theresult from the first stage via a greedy scheme that solves restrictednonconvex quadratically constrained quadratic programs (QCQPs). We show thatthe optimal value of such a QCQP can be obtained by solving their semidefiniterelaxations. Numerical experiments on S\&P 500 are conducted to demonstrate theeffectiveness of the proposed algorithm.

A Penalty Decomposition Algorithm with Greedy Improvement for Mean-Reverting Portfolios with Sparsity and Volatility Constraints