Projected-Search Methods for Bound-Constrained Optimization

Abstract

Projected-search methods for bound-constrained optimization are based on performing a search along a piecewise-linear continuous path obtained by projecting a search direction onto the feasible region. A benefit of these methods is that many changes to the active set can be made at the cost of computing a single search direction. As the objective function is not differentiable along the search path, it is not possible to use a projected-search method with a step that satisfies the Wolfe conditions, which require the directional derivative of the objective function at a point on the path. Thus, methods based on a simple backtracking procedure must be used to give a step that satisfies an "Armijo-like" sufficient decrease condition. As a consequence, conventional projected-search methods are unable to exploit sophisticated safeguarded polynomial interpolation techniques that have been shown to be effective for the unconstrained case. This paper concerns the formulation and analysis of projected-search methods based on a new quasi-Wolfe line search that is appropriate for piecewise differentiable functions. The behavior of the line search is similar to conventional Wolfe line search, except that a step is accepted under a wider range of conditions. These conditions take into consideration steps at which the restriction of the objective function on the search path is not differentiable. Two new classes of method are proposed that may be broadly categorized as active-set and interior-point methods. Computational results are given for two specific methods from these general classes: a projected-search active-set method that uses a limited-memory quasi-Newton approximation of the Hessian; and a projected-search primal-dual interior-point method. The results show that in these contexts, a quasi-Wolfe line search is substantially more efficient and reliable than an Armijo line search.