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

We introduce and study a new class of generalized convex functions termed star quasiconvex functions. This class includes convex, star-convex, quasiconvex, quasar-convex, and positively homogeneous functions of any degree $p>0$ as special cases. Furthermore, we provide several characterizations of this class covering both nonsmooth and differentiable cases. In particular, in the general nonsmooth case, star quasiconvex functions are characterized by functions for which all its sublevel sets are star-shaped at a minimizer, while in the differentiable case strongly star quasiconvex functions coincides with those satisfying the restricted secant inequality property, and star quasiconvex functions are related with variationally coherent functions, thereby providing a rich framework for differentiable first-order methods. Additionally, we develop standard properties of the proximity operator and 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 that are not necessarily convex.