Solving Euclidean Distance Multifacility Location Problems Using Conjugate Subgradient and Line-Search Methods

Abstract

Subgradient methods are popular for solving nondifferentiable optimization problems because of their relative ease in implementation, but are not always robust and require a careful design of strategies in order to yield an effective procedure for any given class of problems. In this paper, we present an approach for solving the Euclidean distance multifacility location problem (EMFLP) using conjugate or deflected subgradient based algorithms along with suitable line-search strategies. The subgradient deflection method considered is the Average Direction Strategy (ADS) imbedded within the Variable Target Value Method (VTVM). We also investigate the generation of two types of subgradients to be employed in conjunction with ADS. The first type is a simple valid subgradient that assigns zero values to contributions corresponding to the nondifferentiable terms in the objective function, and so, the subgradient is composed by summing the contributions corresponding to the differentiable terms alone. The second type expends more effort to derive a low-norm member of the subdifferential in order to enhance the prospect of obtaining a descent direction. Furthermore, a special Newton-based line-search that exploits the nondifferentiability of the problem is also designed to be implemented in the developed algorithm in order to study its impact on the convergence behavior. Various combinations of the above strategies are composed and evaluated on a set of test problems. The results show that a modification of the VTVM method along with the first or a certain combination of the two subgradient generation strategies, and the use of a suitable line-search technique, provides promising results. An alternative block-halving step-size strategy used within VTVM in conjunction with the proposed line-search method yields a competitive second choice performance.