Maximization of a Function with Lipschitz Continuous Gradient

Abstract

In the present paper, we consider (nonconvex in the general case) functions that have Lipschitz continuous gradient. We prove that the level sets of such functions are proximally smooth and obtain an estimate for the constant of proximal smoothness. We prove that the problem of maximization of such function on a strongly convex set has a unique solution if the radius of strong convexity of the set is sufficiently small. The projection algorithm (similar to the gradient projection algorithm for minimization of a convex function on a convex set) for solving the problem of maximization of such a function is proposed. The algorithm converges with the rate of geometric progression.

Gradient Projection and Conditional Gradient Methods for Constrained Nonconvex Minimization

“…But indeed the solutions of these two problems coincide under condition (24). The proof of Theorem 4 follows from the following fact regarding strongly convex sets of radius r and functions with the Lipschitz continuous gradient [3].…”

“…As an example we consider a quadratic form on the unit sphere. (3) For approximately linear objective functions we propose a new version of the Frank-Wolfe (conditional gradient) method and establish its linear convergence to a global minimum in problem (1). We prove linear convergence of the method for a surface, which is the boundary of a strongly convex set.…”

Gradient Methods on Strongly Convex Feasible Sets and Optimal Control of Affine Systems

“…If f is only convex (but not necessarily strongly convex), the sequence {w k } converges weakly [20]. When K is strongly convex, the linear convergence of {w k } is obtained under additional conditions (too strong for our main application) in [5,6,15].…”