A variational inequality approach for constrained multifacility Weber problem under gauge

Abstract

The classical multifacility Weber problem (MFWP) is one of the most important models in facility location. This paper considers more general and practical case of MFWP called constrained multifacility Weber problem (CMFWP), in which the gauge is used to measure distances and locational constraints are imposed to facilities. In particular, we develop a variational inequality approach for solving it. The CMFWP is reformulated into a linear variational inequality, whose special structures lead to new projection-type methods. Global convergence of the projection-type methods is proved under mild assumptions. Some preliminary numerical results are reported which verify the effectiveness of proposed methods.

ADMM-type methods for generalized multi-facility Weber problem

“…4. ADMM-type methods for the primal problem (19). This section aims to develop efficient numerical methods for the primal problem (P) by exploiting its favourable structure.…”

“…Based on the previous discussion on the iterative scheme and the corresponding subproblems, now we are ready to present our first ADMM-type method for (19), denoted as ADMM-a, as follows. Algorithm 1.…”

“…Algorithm 1. ADMM-a for (19) Given the parameter β > 0, r > 0 and initial iterate w 0 = (x 0 , y 0 , λ 0 ).…”

“…Remark 3. The paper [19] considered similar generalized MFWP. However, in [19] a variational inequality (VI) approach was developed and new projection-type methods under VI were proposed for solving GMFWP.…”

“…The paper [19] considered similar generalized MFWP. However, in [19] a variational inequality (VI) approach was developed and new projection-type methods under VI were proposed for solving GMFWP. This paper reformulates GMFWP as a primal problem of the equivalent min-max problem and will propose ADMM-type methods to solve it.…”