金融优化中非凸优化模型理论与算法研究

This project aims to systematically investigate the theory and algorithms for portfolio optimization models with nonconvex constraints. Based on different risk measures and transaction rules, these constraints capture the portfolio risk more accurately and make the portfolio more applicable in practical investments. In this project, we will respectively study the portfolio optimization models with parameter sensitivity control, Value-at-Risk constraint, sparsity constraint and minimum threshold constraint, which are all nonconvex programming problems. These models mostly can be reformulated as quadratic programs with D.C. constraints and complementary constraints. We will try to explore the methods for these models and figure out the theoretical features of the optimal solutions mainly via the theory and algorithms of D.C. program and complementary problem. Furthermore, we will carry on the numerical experiment and empirical analysis using the real financial data.

本项目旨在研究带非凸约束的风险管理和投资组合模型的相关理论与算法。基于不同的风险度量方法和交易规则的限制，这些非凸约束更好刻画了投资组合的风险特征，且使得投资策略更符合实际投资特征。在本项目中将分别研究带参数敏感度约束的均值方差模型、带风险值约束的投资模型、带有稀疏性要求的投资模型以及带有最小持有量约束的投资组合模型。这些金融优化模型均属于非凸优化问题，大多等价于带D.C.约束的二次规划问题或互补问题。我们将利用对D.C.规划问题以及互补问题的理论与算法对这些模型的求解方法以及最优解所具有的特点进行深入探讨和分析，并利用国内外真实数据对这些模型进行数值试验和实证分析。

本项目主要研究了基于不同风险度量和约束条件的投资模型，分别对带参数敏感度约束的投资模型、基于风险值度量的投资模型以及基于极端风险度量的投资模型。由于这些模型均为非凸模型，我们首先从最优化角度对其局部解和全局解的求解进行深入探索，进行大量数值实验，取得了一系列显著成果。我们也利用真实历史数据对模型的合理性和实证表现进行分析。数值实验和实证分析表明，这些模型可为量化投资者提供了有力的决策工具。