Quasar-convex functions admit high-order proximal algorithms with linear convergence for p=2 and superlinear for p>2 under suitable conditions.
Extending Linear Convergence of the Proximal Point Algorithm: The Quasar-Convex Case
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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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math.OC 2years
2026 2verdicts
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Robust learning problems are formulated as quasar-convex optimization, and HiPPA is proposed as an inexact high-order proximal method with global and superlinear convergence guarantees.
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Quasar-Convex Optimization: Fundamental Properties and High-Order Proximal-Point Methods
Quasar-convex functions admit high-order proximal algorithms with linear convergence for p=2 and superlinear for p>2 under suitable conditions.
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Robust Learning Meets Quasar-Convex Optimization: Inexact High-Order Proximal-Point Methods
Robust learning problems are formulated as quasar-convex optimization, and HiPPA is proposed as an inexact high-order proximal method with global and superlinear convergence guarantees.