一种用于锥约束与流形约束复合优化的近端线性化NEP方法
A proximal-linearized NEP method for composite optimization with conic and manifold constraints
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中文总结 AI 辅助
本文提出一种非精确近端线性化精确惩罚算法，用于求解带锥约束与流形约束的DC复合优化问题，在温和假设下达到O(ε^{-2})复杂度，数值实验验证其有效性。
中文摘要 AI 辅助
本文研究具有锥约束与流形约束同时存在的DC复合优化问题。我们提出一种基于距离惩罚函数的非精确近端线性化精确惩罚算法来处理锥约束。该算法在易于验证的非精确性准则下，在流形的切空间上依次计算惩罚函数近端线性化的近似极小点，并自适应更新近端参数和惩罚参数。在迭代序列和惩罚参数序列有界的假设下，我们建立了寻找ε-稳定点的O(ε^{-2})最坏情况复杂度界。此外，当采用半近端ADMM求解子问题时，若距离函数和第一个DC分量是分段线性二次的，则所得预言复杂度为O(ε^{-2}log ε^{-1})，而加速变体可实现O(ε^{-4})的预言复杂度。若进一步满足相关势函数具有KL性质，则整个迭代序列收敛到稳定点。在正交流形、非负锥和二阶锥（SOC）约束的复合优化问题上的大量数值实验证明了所提方法的有效性。
英文摘要
This paper studies difference-of-convex (DC) composite optimization problems with conic and manifold constraints. By penalizing the conic constraint with a distance-based penalty, we propose an inexact proximal-linearized nonsmooth exact penalty (iPLNEP) algorithm. The proposed method successively finds approximate minimizers of proximal-linearized subproblems over the tangent spaces of the manifold based on computable inexactness criteria, while adaptively updating the proximal and penalty parameters. Under a boundedness assumption on the iterate and penalty parameter sequences, iPLNEP is shown to achieve an $O(ε^{-2})$ worst-case iteration complexity bound for finding an $ε$-stationary point. Using accelerated semi-proximal ADMM to solve the subproblems, we establish an overall oracle complexity bound of $O(ε^{-4})$. Under an additional uniform error-bound condition, using semi-proximal ADMM as the subproblem solver yields an improved overall oracle complexity bound of $O(ε^{-2}\logε^{-1})$. To the best of our knowledge, this provides the first proximal-linearized NEP framework with provable oracle complexity guarantees for DC composite optimization with conic and manifold constraints. If, in addition, the associated potential function satisfies the KL property, the whole sequence of iterates converges to a stationary point. Extensive numerical experiments on composite optimization problems with orthogonal-manifold, nonnegative cone, and second-order cone constraints demonstrate the effectiveness of the proposed method.
发表机构
- School of Mathematics, South China University of Technology（华南理工大学数学学院）
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