Block Coordinate Descent for smooth nonconvex constrained minimization

TL;DR

This paper develops a block coordinate descent algorithm for smooth nonconvex constrained problems with complex constraints, proving convergence and complexity, and demonstrating its effectiveness on a continuous traveling salesman problem.

Contribution

It introduces a general algorithm for nonconvex constrained minimization with complex block constraints, providing convergence analysis and complexity bounds.

Findings

Algorithm converges to critical points with complexity $O(rac{1}{\, ext{epsilon}^2})$

Proven convergence under complex, non-simple block constraints

Numerical results on continuous traveling salesman problem demonstrate effectiveness

Abstract

At each iteration of a Block Coordinate Descent method one minimizes an approximation of the objective function with respect to a generally small set of variables subject to constraints in which these variables are involved. The unconstrained case and the case in which the constraints are simple were analyzed in the recent literature. In this paper we address the problem in which block constraints are not simple and, moreover, the case in which they are not defined by global sets of equations and inequations. A general algorithm that minimizes quadratic models with quadratric regularization over blocks of variables is defined and convergence and complexity are proved. In particular, given tolerances $\delta>0$ and $\varepsilon>0$ for feasibility/complementarity and optimality, respectively, it is shown that a measure of $(\delta,0)$-criticality tends to zero; and the the number of iterations and functional evaluations required to achieve $(\delta,\varepsilon)$-criticality is $O(\varepsilon^2)$. Numerical experiments in which the proposed method is used to solve a continuous version of the traveling salesman problem are presented.