  Mean-reverting portfolios with few assets, but high variance, are of great
interest for investors in financial markets. Such portfolios are
straightforwardly profitable because they include a small number of assets
whose prices not only oscillate predictably around a long-term mean but also
possess enough volatility. Roughly speaking, sparsity minimizes trading costs,
volatility provides arbitrage opportunities, and mean-reversion property equips
investors with ideal investment strategies. Finding such favorable portfolios
can be formulated as a nonconvex quadratic optimization problem with an
additional sparsity constraint. To the best of our knowledge, there is no
method for solving this problem and enjoying favorable theoretical properties
yet. In this paper, we develop an effective two-stage algorithm for this
problem. In the first stage, we apply a tailored penalty decomposition method
for finding a stationary point of this nonconvex problem. For a fixed penalty
parameter, the block coordinate descent method is utilized to find a stationary
point of the associated penalty subproblem. In the second stage, we improve the
result from the first stage via a greedy scheme that solves restricted
nonconvex quadratically constrained quadratic programs (QCQPs). We show that
the optimal value of such a QCQP can be obtained by solving their semidefinite
relaxations. Numerical experiments on S\&P 500 are conducted to demonstrate the
effectiveness of the proposed algorithm.
