二次极小极大优化的单循环方法
A single loop method for quadratic minmax optimization
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中文总结 AI 辅助
针对带耦合内部约束的二次极小极大问题，提出一种不区分内外层的不可行内点型单循环方法，证明其非退化平稳点的吸引性，且在特定情形下具多项式复杂度，实验显示其优于现有算法。
中文摘要 AI 辅助
我们考虑带有耦合内部约束的二次极小极大问题，并提出一种计算一类平稳点的方法。为说明计算此类平稳点的必要性，我们首先证明其具有实际意义，即它们在合适的非退化条件下可成为该问题的局部最优解。随后，基于合适的对数障碍函数，我们构建了一种不可行内点型单循环方法（该方法不明确区分外层与内层问题），并证明当算法沿设计的中心路径移动时，非退化平稳点是吸引点。我们特别指出，在带约束极小极大问题的内层可行集与外层变量无关的特殊情形下，该方法具有多项式复杂度。我们在合成数据和一类最小费用流问题上开展数值实验，展示了该方法的性能，以及它在计算平稳点的质量方面如何优于现有文献中的算法。
英文摘要
We consider a quadratic minmax problem with coupled inner constraints and propose a method to compute a class of stationary points. To motivate the need to compute such stationary points, we first show that they are meaningful, in the sense that they can be locally optimal for our problem under suitable{non-degeneracy} conditions. Then based on a suitable log barrier function, we build an infeasible interior point-type {single loop method} (which does not explicitly distinguish between the outer and inner problem) and prove that a non-degenerate stationary point is an attraction point as the algorithm moves along the designed central path. We show in particular that our method is polynomial in the special case where the inner feasible set of our constrained minmax problem is independent from outer variables. Our numerical experiments, on both synthetic data and a class of min-cost flow problems, showcase the behavior of our method and how it outperforms existing algorithms from the literature in terms of the quality of the computed stationary points.
