This paper studies mean-risk portfolio optimization models using the conditional value-at-risk (CVaR) as a risk measure. We also employ a cardinality constraint for limiting the number of invested assets. Solving such a cardinality-constrained mean-CVaR model is computationally challenging for two main reasons. First, this model is formulated as a mixed-integer optimization (MIO) problem because of the cardinality constraint, so solving it exactly is very hard when the number of investable assets is large. Second, the problem size depends on the number of asset return scenarios, and the computational efficiency decreases when the number of scenarios is large. To overcome these challenges, we propose a high-performance algorithm named the bilevel cutting-plane algorithm for exactly solving the cardinality-constrained mean-CVaR portfolio optimization problem. We begin by reformulating the problem as a bilevel optimization problem and then develop a cutting-plane algorithm for solving the upper-level problem. To speed up computations for cut generation, we apply to the lower-level problem another cutting-plane algorithm for efficiently minimizing CVaR with a large number of scenarios. Moreover, we prove the convergence properties of our bilevel cutting-plane algorithm. Numerical experiments demonstrate that, compared with other MIO approaches, our algorithm can provide optimal solutions to large problem instances faster.

本文研究了使用条件风险价值 (CVaR) 作为风险度量的平均风险投资组合优化模型。我们还使用基数约束来限制投资资产的数量。由于两个主要原因，解决这种基数约束的均值 CVaR 模型在计算上具有挑战性。首先，由于基数约束，该模型被表述为混合整数优化 (MIO) 问题，因此当可投资资产数量较大时，很难准确求解。其次，问题的大小取决于资产回报场景的数量，场景数量多时计算效率降低。为了克服这些挑战，我们提出了一种名为双层切割平面算法的高性能算法用于精确解决基数约束均值 CVaR 投资组合优化问题。我们首先将问题重新表述为双层优化问题，然后开发用于解决上层问题的切割平面算法。为了加速切割生成的计算，我们将另一种切割平面算法应用于较低级别的问题，以有效地最小化具有大量场景的 CVaR。此外，我们证明了我们的双层切割平面算法的收敛性。数值实验表明，与其他 MIO 方法相比，我们的算法可以更快地为大型问题实例提供最佳解决方案。