Dmitriy Drusvyatskiy

Tilt Stability, Uniform Quadratic Growth, and Strong Metric Regularity of the Subdifferential

We prove that uniform second-order growth, tilt stability, and strong metric regularity of the subdifferential---three notions that have appeared in entirely different settings---are all essentially equivalent for any lower-semicontinuous, extended real-valued function.

Transversality and Alternating Projections for Nonconvex Sets

We consider the method of alternating projections for finding a point in the intersection of two closed sets, possibly nonconvex. Assuming only the standard transversality condition (or a weaker version thereof), we prove local linear convergence. When the two sets are semi-algebraic and bounded, but not necessarily transversal, we nonetheless prove subsequence convergence.

Efficiency of minimizing compositions of convex functions and smooth maps

We consider global efficiency of algorithms for minimizing a sum of a convex function and a composition of a Lipschitz convex function with a smooth map. The basic algorithm we rely on is the prox-linear method, which in each iteration solves a regularized subproblem formed by linearizing the smooth map. When the subproblems are solved exactly, the method has efficiency

Coordinate Shadows of Semidefinite and Euclidean Distance Matrices

\mathcal {O}(\varepsilon ^{-2})

Semi-algebraic functions have small subdifferentials

O(ε-2), akin to gradient descent for smooth minimization. We show that when the subproblems can only be solved by first-order methods, a simple combination of smoothing, the prox-linear method, and a fast-gradient scheme yields an algorithm with complexity

An Optimal First Order Method Based on Optimal Quadratic Averaging

\widetilde{\mathcal {O}}(\varepsilon ^{-3})