A generalized Weber problem with different gauges for different regions

This paper considers a generalized variation of the Weber problem (GVWP) on the plane in which a straight line divides the plane into two regions and different gauges are employed to measure the distances in different regions. GVWP is a generalized problem of some well-studied variations of Weber problem (VWP) in the following aspects: (1) both the unconstrained and constrained problems are taken into consideration; (2) the more general distance measuring functions, gauges  , are employed to measure distances instead of the usually used lplp-norms. Therefore, the GVWP is more practical and applicable in practice. GVWP is nonconvex as VWP, and as a generalized problem it is more complex than the latter. In this paper the GVWP is divided into three subproblems which are optimally solved: two subproblems are reformulated into monotone linear variational inequalities (LVIs) and then solved by a projection–contraction method, while the third subproblem is divided into some convex problems and the golden section search is used to solve them. By dividing the GVWP into three subproblems and solving these subproblems optimally, an algorithm which can obtain the global optimal solution of GVWP is proposed. Preliminary numerical results are reported to verify the evident effectiveness of the proposed algorithm.