{"ID":2874707,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.04375","arxiv_id":"2509.04375","title":"Extending Linear Convergence of the Proximal Point Algorithm: The Quasar-Convex Case","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.","short_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...","url_abs":"https://arxiv.org/abs/2509.04375","url_pdf":"https://arxiv.org/pdf/2509.04375v2","authors":"[\"José de Brito\",\"Felipe Lara\",\"Di Liu\"]","published":"2025-09-04T16:32:43Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[]","has_code":false}