发表机构
HEC Montréal; Instituto de Alta Investigación (IAI), Universidad de Tarapacá（蒙特利尔高等商学院; 塔拉帕卡大学高级研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出弱类拟凸函数概念，建立其理论基础，分析邻近点算法的收敛性与复杂度，构造出不属于现有各类凸性框架的弱类拟凸函数，拓展了凸优化的适用范围。
AI 中文摘要
本文引入弱类拟凸函数这一类函数，该概念同时推广了弱凸性（进而凸性）和类拟凸性。我们通过建立其基本运算规则、推导其特征并系统研究其闭包与结构性质，为这一新类函数奠定理论基础。基于这些工具，我们分析了用于最小化弱类拟凸函数的邻近点算法，该函数满足局部二次增长条件，证明了迭代序列的Q-线性收敛性以及函数值到全局极小点的R-线性收敛性，迭代复杂度为O(ln(ε⁻¹))；我们还通过邻近残差给出了一种完全可计算、无需极小点的停止准则，其复杂度与前者匹配。最后，我们构造了一个满足所有既定假设的弱类拟凸函数，该函数既非弱凸、拟凸、星型拟凸，也非任意模的类拟凸，表明我们的收敛理论适用于文献中所有现有框架之外的实例。
英文摘要
This paper introduces the class of weakly quasar-convex functions, a notion that simultaneously generalizes weak convexity (and hence convexity) and quasar-convexity. We establish the theoretical foundation for this new class by developing its basic calculus, deriving characterizations, and systematically investigating its closure and structural properties. Building on these tools, we analyze the proximal point algorithm for minimizing a weakly quasar-convex function under local quadratic growth, proving $Q$-linear convergence of the iterates and $R$-linear convergence of the function values to a global minimizer, with iteration complexity $O(\ln(\varepsilon^{-1}))$; we also give a fully computable, minimizer-free stopping rule via the proximal residual with matching complexity. Finally, we construct a weakly quasar-convex function, satisfying all our standing assumptions, that is neither weakly convex, quasi-convex, star quasi-convex, nor quasar-convex for any modulus, showing that our convergence theory applies to instances outside every prior framework in the literature.
