A convergent decomposition method for box-constrained optimization problems.

In this work we consider the problem of minimizing a continuously differentiable function over a feasible set defined by box constraints. We present a decomposition method based on the solution of a sequence of subproblems. In particular, we state conditions on the rule for selecting the subproblem variables sufficient to ensure the global convergence of the generated sequence without convexity assumptions. The conditions require to select suitable variables (related to the violation of the optimality conditions) to guarantee theoretical convergence properties, and leave the degree of freedom of selecting any other group of variables to accelerate the convergence. © 2009 Springer-Verlag

Forward-backward truncated Newton methods for convex composite optimization

This paper proposes two proximal Newton-CG methods for convex nonsmooth optimization problems in composite form. The algorithms are based on a a reformulation of the original nonsmooth problem as the unconstrained minimization of a continuously differentiable function, namely the forward-backward envelope (FBE). The first algorithm is based on a standard line search strategy, whereas the second one combines the global efficiency estimates of the corresponding first-order methods, while achieving fast asymptotic convergence rates. Furthermore, they are computationally attractive since each Newton iteration requires the approximate solution of a linear system of usually small dimension

A decomposition procedure based on approximate newton directions

The efficient solution of large-scale linear and nonlinear optimization problems may require exploiting any special structure in them in an efficient manner. We describe and analyze some cases in which this special structure can be used with very little cost to obtain search directions from decomposed subproblems. We also study how to correct these directions using (decomposable) preconditioned conjugate gradient methods to ensure local convergence in all cases. The choice of appropriate preconditioners results in a natural manner from the structure in the problem. Finally, we conduct computational experiments to compare the resulting procedures with direct methods, as well as to study the impact of different preconditioner choices