基于幂和DC分解的Boosted-DCA求解线性约束多项式规划
A Boosted-DCA with Power-Sum-DC Decomposition for Linearly Constrained Polynomial Programs
AI总结：
本文提出基于幂和表示的多项式DC分解，并引入BDCAe算法结合FDPG方法高效求解线性约束多项式规划，理论证明其收敛性且数值实验验证了其优越性。
摘要：
本文提出了一种使用幂和表示的多项式新型Difference-of-Convex (DC) 分解，该分解通过求解稀疏线性系统实现。我们引入了带有精确线搜索的Boosted DCA (BDCAe)，用于解决DC框架内的线性约束多项式规划。值得注意的是，我们证明了精确线搜索等价于确定区间内一元多项式的根，其系数基于幂和DC分解显式计算。我们证明了BDCAe到临界点的子序列收敛性，并建立了其在Kurdyka-Lojasiewicz性质下的收敛速率。为了高效解决凸子问题，我们利用幂和DC分解的可分块结构，集成了Fast Dual Proximal Gradient (FDPG) 方法。我们通过在Mean-Variance-Skewness-Kurtosis (MVSK) 投资组合优化模型和箱约束多项式优化问题上的数值实验验证了我们的方法。将BDCAe与DCA、带有Armijo线搜索的BDCA、UDCA、带有投影DC分解的UBDCA，以及标准非线性优化求解器FMINCON和FILTERSD进行比较分析，证实了我们提出方法的效率。
英文摘要：
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.
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