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arXiv · 2509.04375

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

José de Brito·Felipe Lara·Di Liu
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Abstract

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.

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José de Brito, Felipe Lara, Di Liu. 2026-04-14. Extending Linear Convergence of the Proximal Point Algorithm: The Quasar-Convex Case. https://arxiv.org/abs/2509.04375

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