Block coordinate descent for smooth nonconvex constrained minimization

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 quadratic 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 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.