论文标题
带有功率-SUM-DC分解的Boosted-DCA，用于线性约束多项式程序
A Boosted-DCA with Power-Sum-DC Decomposition for Linearly Constrained Polynomial Programs
论文作者
论文摘要
本文提出了通过求解稀疏的线性系统实现的多项式代表的多项式分解差异差异（DC）分解。我们使用精确的线路搜索（BDCAE）介绍了增强的DCA，以解决DC框架中线性约束多项式程序的解决。值得注意的是，我们证明了确切的线搜索等同于在间隔中确定单变量多项式的根，并且根据功率-SUM DC分解明确计算系数。 BDCAE与关键点的随后收敛已被证明，并且在Kurdyka-Lojasiewicz属性下的收敛速率也得到了证明。为了有效地解决凸子问题，我们通过利用功率-SUM DC分解的可分离块结构来整合快速双重近端梯度（FDPG）方法。我们通过数值实验验证了我们的方法，这些实验对平均变化 - 扎实 - 库尔图病（MVSK）投资组合优化模型和盒子约束的多项式优化问题。 BDCAE针对DCA，BDCA与Armijo线搜索，UDCA和UBDCA的比较分析，以及投影DC分解，以及标准的非线性优化求解器Fmincon和FilterSD，证明了我们建议的方法的效率。
This paper proposes a novel Difference-of-Convex (DC) decomposition for polynomials using a power-sum representation, achieved by solving a sparse linear system. We introduce the Boosted DCA with Exact Line Search (BDCAe) for addressing linearly constrained polynomial programs within the DC framework. Notably, we demonstrate that the exact line search equates to determining the roots of a univariate polynomial in an interval, with coefficients being computed explicitly based on the power-sum DC decompositions. The subsequential convergence of BDCAe to critical points is proven, and its convergence rate under the Kurdyka-Lojasiewicz property is established. To efficiently tackle the convex subproblems, we integrate the Fast Dual Proximal Gradient (FDPG) method by exploiting the separable block structure of the power-sum DC decompositions. We validate our approach through numerical experiments on the Mean-Variance-Skewness-Kurtosis (MVSK) portfolio optimization model and box-constrained polynomial optimization problems. Comparative analysis of BDCAe against DCA, BDCA with Armijo line search, UDCA, and UBDCA with projective DC decomposition, alongside standard nonlinear optimization solvers FMINCON and FILTERSD, substantiates the efficiency of our proposed approach.
