Inexact Primal-Dual Gradient Projection Methods for Nonlinear Optimization on Convex Set

TL;DR

This paper introduces an inexact primal-dual gradient projection method for nonlinear optimization with convex constraints, reducing computational costs while maintaining convergence guarantees, especially effective for l1-ball constrained problems.

Contribution

It presents a novel inexact gradient projection approach with proven convergence rates, applicable to problems with difficult projection computations.

Findings

Method reduces projection computation cost.

Achieves global convergence with O(1/k) rate.

More efficient than exact methods for l1-ball problems.

Abstract

In this paper, we propose a novel primal-dual inexact gradient projection method for nonlinear optimization problems with convex-set constraint. This method only needs inexact computation of the projections onto the convex set for each iteration, consequently reducing the computational cost for projections per iteration. This feature is attractive especially for solving problems where the projections are computationally not easy to calculate. Global convergence guarantee and O(1/k) ergodic convergence rate of the optimality residual are provided under loose assumptions. We apply our proposed strategy to l1-ball constrained problems. Numerical results exhibit that our inexact gradient projection methods for solving l1-ball constrained problems are more efficient than the exact methods.