The multi-source Weber problem (MWP) is a very important model in facility location and Cooper algorithm is the most well-used method to solve MWP. Cooper algorithm consists of two steps:location step and allocation step. By doing such two steps alternately, the local optimal solution can be obtained. In this paper, we propose a modified Cooper algorithm by introducing improved strategy for both steps. For location step, we combine the Weiszfeld method with the adaptive Barzilai-Borwein (ABB) method to propose ABBWeiszfeld method, which has a faster convergence rate in solving location subproblems. For allocation step, we propose a greedy cluster splitting strategy to deal with out-of-use facilities, and then we propose a mixed greedy strategy which has better properties. Some preliminary numerical experiments are reported which have shown that the proposed strategies improve the computational efficiency of Cooper algorithm and thus result in a modified algorithm which has better performance.
Fang X W. A direct search frame-based adaptive Barzilai-Borwein method[J]. Journal of Computational Mathematics, 2015, 33(2):179-190.
Zanghirati G, Zanni L, Frassoldati G. New adaptive stepsize selections in gradient methods[J]. Journal of Industrial & Management Optimization, 2017, 4(2):299-312.
Brimberg J, Mladenovic N. Degeneracy in the multi-source Weber problem[J]. Mathematical Programming, 1999, 85(1):213-220.
Zanjirani F R, Hekmatfar M. Facility location:concepts, models, algorithms and case studies[J]. Applications and theory, 2009, 28(1):65-81.
Weiszfeld E. Sur le point pour lequel la somme des distances de n points donn es est minimum[J]. Faseb Journal Official Publication of the Federation of American Societies for Experimental Biology, 2006, 20(3):559-566.
Vardi Y, Zhang C H. A modified Weiszfeld algorithm for the Fermat-Weber location problem[J]. Mathematical Programming, 2001, 90(3):559-566.
Barazilai J, Borwein J M. Two-point step size gradient methods[J]. IMA Journal of Numerical Analysis, 1988, 8(1):141-148.
Fletcher R. Low storage methods for unconstrained optimization[M]. Lectures in Applied Mathematics, 1990, 26:165-179.
Andreani R, Birgin E G, Mart nez J M, et al. Spectral projected gradient and variable metric methods for optimization with linear inequalities[J]. IMA Journal of Numerical Analysis, 2005, 25(2):221-252.
Lenys B, Marcos R. Preconditioned spectral projected gradient method on convex sets[J]. Journal of Computational Mathematics, 2005, 23(3):225-232.
Birgin E G, Martinez J M, Raydan M. Nonmonotone spectral projected gradient methods for convex sets[J]. IMA Journal of Numerical Analysis, 2003, 23(23):539-559.
Zhang H C, Hager W W. A nonmonotone line search technique and its application to unconstrained optimization[J]. SIAM Journal on Optimization, 2006, 14(4):1043-1056.
Zhou B, Gao L, Dai Y H. Gradient methods with adaptive step-sizes[J]. Computational Optimization & Applications, 2006, 35(1):69-86.
Kuhn H W. A note on Fermat's problem[J]. Mathematical Programming, 1973, 4(1):98-107.
Jiang J L, Yuan X M. A heuristic algorithm for constrained multi-source Weber problem - The variational inequality approach[J]. European Journal of Operational Research, 2008, 187(2):357-370.