An almost cyclic 2-coordinate descent method for singly linearly constrained problems

TL;DR

This paper introduces an almost cyclic 2-coordinate descent method for constrained optimization that avoids full gradient computations, guarantees convergence to stationary points, and demonstrates efficiency through numerical experiments.

Contribution

It proposes a novel coordinate selection strategy that does not rely on first-order information, with proven convergence and linear rate under standard assumptions.

Findings

Global convergence to stationary points is established.

Linear convergence rate is proved under additional assumptions.

Numerical results confirm the method's effectiveness.

Abstract

A block decomposition method is proposed for minimizing a (possibly non-convex) continuously differentiable function subject to one linear equality constraint and simple bounds on the variables. The proposed method iteratively selects a pair of coordinates according to an almost cyclic strategy that does not use first-order information, allowing us not to compute the whole gradient of the objective function during the algorithm. Using first-order search directions to update each pair of coordinates, global convergence to stationary points is established for different choices of the stepsize under an appropriate assumption on the level set. In particular, both inexact and exact line search strategies are analyzed. Further, linear convergence rate is proved under standard additional assumptions. Numerical results are finally provided to show the effectiveness of the proposed method.