非光滑非凸–强拟凹鞍点问题的Bregman近端梯度算法
Bregman Proximal Gradient Algorithm for Nonsmooth Nonconvex-Strongly Quasi-Concave Saddle Point Problems
摘要: 针对非光滑非凸–强拟凹鞍点问题，本文利用Bregman距离建立了Bregman近端梯度上升下降算法。对Bregman近端梯度上升迭代算法中，得到内部最大化问题函数差值不等式，从而得到近端梯度上升迭代点之间的不等式关系。对于非凸非光滑问题，引入扰动类梯度下降序列，得到算法的次收敛性，当目标函数为半代数时，得到算法的全局收敛性。
Abstract: For the nonsmooth nonconvex-strongly quasi-concave saddle point problems, this paper establishes the Bregman proximal gradient ascent-descent algorithm by using the Bregman distance. In the Bregman proximal gradient ascent iterative algorithm, the difference inequality of the internal maximization problem function is obtained, and thus the inequality relationship between the proxi-mal gradient ascent iterative points is derived. For nonconvex and nonsmooth problems, a perturbed gradient-like descent sequence is introduced to obtain the sub-convergence of the algorithm. When the objective function is semi-algebraic, the global convergence of the algorithm is obtained.
文章引用：张艳, 李小兵. 非光滑非凸–强拟凹鞍点问题的Bregman近端梯度算法[J]. 应用数学进展, 2025, 14(1): 442-452. https://doi.org/10.12677/aam.2025.141043

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