An ADMM-based location–allocation algorithm for nonconvex constrained multi-source Weber problem under gauge

Multi-source Weber problem (MSWP) is a classical nonconvex and NP-hard model in facility location. A well-known method for solving MSWP is the location–allocation algorithm which consists of a location phase to locate new facilities and an allocation phase to allocate customers at each iteration. This paper considers the more general and practical case of MSWP called the constrained multi-source Weber problem (CMSWP), i.e., locating multiple facilities with the consideration of the gauge for measuring distances and locational constraints on new facilities. According to the favorable structure of the involved location subproblems after reformulation, an alternating direction method of multipliers (ADMM) type method is contributed to solving these subproblems under different distance measures in a uniform framework. Then a new ADMM-based location–allocation algorithm is presented for CMSWP and its local convergence is theoretically proved. Some preliminary numerical results are reported to verify the effectiveness of proposed methods.

[1]  Bingsheng He,et al.  The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent , 2014, Mathematical Programming.

[2]  Keinosuke Fukunaga,et al.  Introduction to Statistical Pattern Recognition , 1972 .

[3]  Shiqian Ma,et al.  On the Nonergodic Convergence Rate of an Inexact Augmented Lagrangian Framework for Composite Convex Programming , 2016, Math. Oper. Res..

[4]  M. Hestenes.  Multiplier and gradient methods , 1969 .

[5]  M. Shirosaki.  Another proof of the defect relation for moving targets , 1991 .

[6]  Frank Plastria,et al.  Asymmetric distances, semidirected networks and majority in Fermat–Weber problems , 2009, Ann. Oper. Res..

[7]  B. Mercier,et al.  A dual algorithm for the solution of nonlinear variational problems via finite element approximation , 1976 .

[8]  P. Lions,et al.  Splitting Algorithms for the Sum of Two Nonlinear Operators , 1979 .

[9]  Xiaoming Yuan,et al.  A heuristic algorithm for constrained multi-source Weber problem - The variational inequality approach , 2008, Eur. J. Oper. Res..

[10]  Leon Cooper,et al.  Heuristic Methods for Location-Allocation Problems , 1964 .

[11]  Leon Cooper,et al.  SOLUTIONS OF GENERALIZED LOCATIONAL EQUILIBRIUM MODELS , 1967 .

[12]  Adi Ben-Israel,et al.  A heuristic method for large-scale multi-facility location problems , 2004, Comput. Oper. Res..

[13]  Wotao Yin,et al.  On the Global and Linear Convergence of the Generalized Alternating Direction Method of Multipliers , 2016, J. Sci. Comput..

[14]  Nimrod Megiddo,et al.  On the Complexity of Some Common Geometric Location Problems , 1984, SIAM J. Comput..

[15]  Stephen P. Boyd,et al.  Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers , 2011, Found. Trends Mach. Learn..

[16]  Bingsheng He,et al.  Linearized Alternating Direction Method with Gaussian Back Substitution for Separable Convex Programming , 2011 .

[17]  Loo Hay Lee,et al.  Multi-source facility location-allocation and inventory problem , 2010, Eur. J. Oper. Res..

[18]  Michael Elad,et al.  From Sparse Solutions of Systems of Equations to Sparse Modeling of Signals and Images , 2009, SIAM Rev..

[19]  Lawrence M. Ostresh.  On the Convergence of a Class of Iterative Methods for Solving the Weber Location Problem , 1978, Oper. Res..

[20]  Xiaoming Yuan,et al.  Recovering Low-Rank and Sparse Components of Matrices from Incomplete and Noisy Observations , 2011, SIAM J. Optim..

[21]  Pierre Hansen,et al.  Improvement and Comparison of Heuristics for Solving the Uncapacitated Multisource Weber Problem , 2000, Oper. Res..

[22]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[23]  Junfeng Yang,et al.  A Proximal Point Algorithm for Log-Determinant Optimization with Group Lasso Regularization , 2013, SIAM J. Optim..

[24]  Dimitri P. Bertsekas,et al.  Constrained Optimization and Lagrange Multiplier Methods , 1982 .

[25]  Bingsheng He,et al.  Some convergence properties of a method of multipliers for linearly constrained monotone variational inequalities , 1998, Oper. Res. Lett..

[26]  Patrick L. Combettes,et al.  Signal Recovery by Proximal Forward-Backward Splitting , 2005, Multiscale Model. Simul..

[27]  H. H. Rachford,et al.  On the numerical solution of heat conduction problems in two and three space variables , 1956 .

[28]  James E. Ward,et al.  Using Block Norms for Location Modeling , 1985, Oper. Res..

[29]  Cun-Hui Zhang,et al.  A modified Weiszfeld algorithm for the Fermat-Weber location problem , 2001, Math. Program..

[30]  R. Love,et al.  Mathematical Models of Road Travel Distances , 1979 .

[31]  B. He,et al.  Alternating Direction Method with Self-Adaptive Penalty Parameters for Monotone Variational Inequalities , 2000 .

[32]  Christian Michelot,et al.  A primal-dual algorithm for the fermat-weber problem involving mixed gauges , 1987, Math. Program..

[33]  Kim-Chuan Toh,et al.  A Convergent 3-Block SemiProximal Alternating Direction Method of Multipliers for Conic Programming with 4-Type Constraints , 2014, SIAM J. Optim..

[34]  Xiaoming Yuan,et al.  Convergence Analysis of Douglas-Rachford Splitting Method for "Strongly + Weakly" Convex Programming , 2017, SIAM J. Numer. Anal..

[35]  Cem Iyigun,et al.  A generalized Weiszfeld method for the multi-facility location problem , 2010, Oper. Res. Lett..

[36]  Said Salhi,et al.  Facility Location: A Survey of Applications and Methods , 1996 .

[37]  John Wright,et al.  RASL: Robust Alignment by Sparse and Low-Rank Decomposition for Linearly Correlated Images , 2012, IEEE Trans. Pattern Anal. Mach. Intell..

[38]  Arie Tamir,et al.  Open questions concerning Weiszfeld's algorithm for the Fermat-Weber location problem , 1989, Math. Program..

[39]  I. Norman Katz,et al.  A Weiszfeld algorithm for the solution of an asymmetric extension of the generalized Fermat location problem , 2010, Comput. Math. Appl..

[40]  Justo Puerto,et al.  Simpson Points in Planar Problems with Locational Constraints. The Polyhedral-Gauge Case , 1997, Math. Oper. Res..

[41]  Xiaoming Yuan,et al.  A sequential updating scheme of the Lagrange multiplier for separable convex programming , 2016, Math. Comput..

[42]  Jack Brimberg,et al.  Global Convergence of a Generalized Iterative Procedure for the Minisum Location Problem with lp Distances , 1993, Oper. Res..

[43]  Dimitri P. Bertsekas,et al.  On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators , 1992, Math. Program..

[44]  Harold W. Kuhn,et al.  A note on Fermat's problem , 1973, Math. Program..