标题： 扩展近似点算法的线性收敛性：准凸情况 显示英文标题

标题： Extending Linear Convergence of the Proximal Point Algorithm: The Quasar-Convex Case

摘要： 本工作研究了拟凸函数的邻近算子的性质，并建立了邻近点算法收敛到全局最小值的结果，特别关注其收敛速度。 具体而言，我们证明了：(i) 生成的序列是极小化的，并且对于拟凸函数实现了$\mathcal{O}(\varepsilon^{-1})$的复杂度率；(ii) 在强拟凸性下，序列线性收敛并达到了$\mathcal{O}(\ln(\varepsilon^{-1}))$的复杂度率。 这些结果将已知的从（强）凸到（强）拟凸设置的收敛率进行了扩展。 据我们所知，某些发现甚至对于（强）星凸函数的特殊情况也是新颖的。 数值实验验证了我们的理论结果。 显示英文摘要

摘要： This work investigates the properties of the proximity operator for quasar-convex functions and establishes the convergence of the proximal point algorithm to a global minimizer with a particular focus on its convergence rate. In particular, we demonstrate: (i) the generated sequence is mi\-ni\-mi\-zing and achieves an $\mathcal{O}(\varepsilon^{-1})$ complexity rate for quasar-convex functions; (ii) under strong quasar-convexity, the sequence converges linearly and attains an $\mathcal{O}(\ln(\varepsilon^{-1}))$ complexity rate. These results extend known convergence rates from the (strongly) convex to the (strongly) quasar-convex setting. To the best of our knowledge, some findings are novel even for the special case of (strongly) star-convex functions. Numerical experiments corroborate our theoretical results.