On Optimal Universal First-Order Methods for Minimizing Heterogeneous Sums

This work considers minimizing a convex sum of functions, each with potentially different structure ranging from nonsmooth to smooth, Lipschitz to non-Lipschitz. Nesterov’s universal fast gradient method provides an optimal black-box first-order method for minimizing a single function that takes advantage of any continuity structure present without requiring prior knowledge. In this paper, we show … Read more

Openness, Holder metric regularity and Holder continuity properties of semialgebraic set-valued maps

Given a semialgebraic set-valued map \$F \colon \mathbb{R}^n \rightrightarrows \mathbb{R}^m\$ with closed graph, we show that the map \$F\$ is Holder metrically subregular and that the following conditions are equivalent: (i) \$F\$ is an open map from its domain into its range and the range of \$F\$ is locally closed; (ii) the map \$F\$ is … Read more