提出快速可扩展的稀疏投资组合优化方法，解决传统算法慢的问题。
A Scalable Gradient-Based Optimization Framework for Sparse Minimum-Variance Portfolio Selection
- 通过布尔松弛将离散选择转为连续优化，保持与原问题等价性。
- 用可调参数控制目标函数从凸到凹，逐步收敛到稀疏解。
- 速度远超商用求解器，多数情况结果接近最优，误差极小。
投资组合优化旨在选择资产权重以最小化风险收益目标，如经典最小方差框架中的组合方差。稀疏投资组合选择通过卡迪纳尔性约束扩展：从 $p$ 个资产中仅允许选择 $k$ 个。标准方法将其建模为混合整数二次规划，依赖商业求解器求解，但计算成本随 $k$ 和 $p$ 指数增长，难以处理中等规模问题。本文提出一种快速、可扩展的基于梯度的方法，通过布尔松弛将组合稀疏选择转化为受限连续优化问题，同时在二值点上保持与原问题的等价性。算法引入可调参数，使辅助目标函数从凸逐渐变为凹，从而实现稳定起点，并沿可控路径向稀疏二值解演化。实际应用中，该方法在大多数实例中与商用求解器的资产选择一致，少数情况下差异仅几项资产，且组合方差误差可忽略。
原文摘要 · Abstract (English)
Portfolio optimization involves selecting asset weights to minimize a risk-reward objective, such as the portfolio variance in the classical minimum-variance framework. Sparse portfolio selection extends this by imposing a cardinality constraint: only $k$ assets from a universe of $p$ may be included. The standard approach models this problem as a mixed-integer quadratic program and relies on commercial solvers to find the optimal solution. However, the computational costs of such methods increase exponentially with $k$ and $p$, making them too slow for problems of even moderate size. We propose a fast and scalable gradient-based approach that transforms the combinatorial sparse selection problem into a constrained continuous optimization task via Boolean relaxation, while preserving equivalence with the original problem on the set of binary points. Our algorithm employs a tunable parameter that transmutes the auxiliary objective from a convex to a concave function. This allows a stable convex starting point, followed by a controlled path toward a sparse binary solution as the tuning parameter increases and the objective moves toward concavity. In practice, our method matches commercial solvers in asset selection for most instances and, in rare instances, the solution differs by a few assets whilst showing a negligible error in portfolio variance.
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