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Star Quasiconvexity: an Unified Approach for Linear Convergence of First-Order Methods Beyond Convexity

Phan Quoc Khanh, Felipe Lara

TL;DR

This work introduces star quasiconvexity as a unifying generalized convexity concept that guarantees linear convergence of first-order methods beyond convexity. It develops multiple equivalent characterizations, including star-shaped sublevel sets and ray-based properties, and derives both differentiable and nonsmooth implications such as quadratic growth and Polyak-Łojasiewicz behavior. The authors prove linear convergence for gradient-type methods (including heavy-ball and Nesterov accelerations) and for the proximal point algorithm on star-shaped domains when strong star quasiconvexity holds with modulus $\gamma>0$. These results extend linear convergence guarantees to a broad nonconvex setting and open avenues for algorithm design on nonconvex problems with star-shaped landscapes. The framework also provides inclusion relations with existing generalized convexities and suggests future work on splitting-type methods and connections to economic reference-point theories.

Abstract

We introduce a class of generalized convex functions, termed star quasiconvexity, to ensure the linear convergence of gradient and proximal point methods. This class encompasses convex, star-convex, quasiconvex, and quasar-convex functions. We establish that a function is star quasiconvex if and only if all its sublevel sets are star-shaped with respect to the set of its minimizers. Furthermore, we provide several characterizations of this class, including nonsmooth and differentiable cases, and derive key properties that fa\-ci\-li\-ta\-te the implementation of first-order methods. Finally, we prove that the proximal point algorithm converges linearly to the unique solution when applied to strongly star quasiconvex functions defined over closed, star-shaped sets, which are not necessarily convex.

Star Quasiconvexity: an Unified Approach for Linear Convergence of First-Order Methods Beyond Convexity

TL;DR

This work introduces star quasiconvexity as a unifying generalized convexity concept that guarantees linear convergence of first-order methods beyond convexity. It develops multiple equivalent characterizations, including star-shaped sublevel sets and ray-based properties, and derives both differentiable and nonsmooth implications such as quadratic growth and Polyak-Łojasiewicz behavior. The authors prove linear convergence for gradient-type methods (including heavy-ball and Nesterov accelerations) and for the proximal point algorithm on star-shaped domains when strong star quasiconvexity holds with modulus . These results extend linear convergence guarantees to a broad nonconvex setting and open avenues for algorithm design on nonconvex problems with star-shaped landscapes. The framework also provides inclusion relations with existing generalized convexities and suggests future work on splitting-type methods and connections to economic reference-point theories.

Abstract

We introduce a class of generalized convex functions, termed star quasiconvexity, to ensure the linear convergence of gradient and proximal point methods. This class encompasses convex, star-convex, quasiconvex, and quasar-convex functions. We establish that a function is star quasiconvex if and only if all its sublevel sets are star-shaped with respect to the set of its minimizers. Furthermore, we provide several characterizations of this class, including nonsmooth and differentiable cases, and derive key properties that fa\-ci\-li\-ta\-te the implementation of first-order methods. Finally, we prove that the proximal point algorithm converges linearly to the unique solution when applied to strongly star quasiconvex functions defined over closed, star-shaped sets, which are not necessarily convex.

Paper Structure

This paper contains 9 sections, 22 theorems, 63 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. An Unifying Approach for Generalized Convex Functions
  4. Motivation
  5. Star Quasiconvexity
  6. The Differentiable Case
  7. Linear Convergence of Gradient-type Me-thods
  8. Linear Convergence of PPA on Star-Shaped Sets
  9. Conclusions and Future Work

Key Result

Lemma 1

(Lara-9) Let $K \subseteq \mathbb{R}^{n}$ be a closed and convex set and $h: \mathbb{R}^{n} \rightarrow \overline{\mathbb{R}}$ be a proper, lsc, and strongly qua-si-con-vex function on $K \subseteq {\rm dom}\,h$ with modulus $\gamma> 0$. Then, ${\rm argmin}_{K} h$ is a singleton.

Figures (3)

  • Figure 1: An illustration of the function $\phi$ in Example \ref{['4:leaf']} (left) and its 4-leaf clover sublevel sets (right).
  • Figure 2: Function $h$ in Example \ref{['exam:02']} when $\alpha = 0.3$ and $k=2$. A 3D plot of $h$ (left) and an arbitrary segment that does not contain the minimizer (right).
  • Figure 3: An illustration of the sublevel sets at height $\delta = 2$ (left) and $\delta = 5$ (right) of function $h$ des-cri-bed in Example \ref{['exam:02']} when $\alpha = 0.3$ and $k=2$.

Theorems & Definitions (51)

  • Lemma 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Proposition 5
  • Remark 6
  • Proposition 7
  • proof
  • Example 8
  • Theorem 9
  • ...and 41 more