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

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.