FAMILY OF PROJECTED DESCENT METHODS FOR OPTIMIZATION PROBLEMS WITH SIMPLE BOUNDS

This paper presents a family of projected descent direction algorithms with inexact line search for solving large-scale minimization problem s subject to simple bounds on the decision variables. The global conve rgence of algorithms in this family is ensured by conditions on the de scent directions and line search. Whenever a sequence constructed by a n algorithm in this family enters a sufficiently small neighborhood of a local minimizer (x) over cap satisfying standard second-order suffi ciency conditions, it gets trapped and converges to this local minimiz er. Furthermore, in this case, the active constraint set at (x) over c ap is identified in a finite number of iterations. This fact is used t o ensure that the rate of convergence to a local minimizer, satisfying standard second-order sufficiency conditions, depends only on the beh avior of the algorithm in the unconstrained subspace. As a particular example, we present projected versions of the modified Polak-Ribiere c onjugate gradient method and the limited-memory BFGS quasi-Newton meth od that retain the convergence properties associated with those algori thms applied to unconstrained problems.