Solving a continuous multifacility location problem by DC algorithms

ABSTRACT

The paper presents a new approach to solve multifacility location problems, which is based on mixed integer programming and algorithms for minimizing differences of convex (DC) functions. The main challenges for solving the multifacility location problems under consideration come from their intrinsic discrete, nonconvex, and nondifferentiable nature. We provide a reformulation of these problems as those of continuous optimization and then develop a new DC type algorithm for their solutions involving Nesterov's smoothing. The proposed algorithm is computationally implemented via MATLAB numerical tests on both artificial and real data sets.

KEYWORDS:

• Mixed integer programming
• multifacility location
• difference of convex functions
• Nesterov's smoothing
• the DCA

AMS SUBJECT CLASSIFICATIONS:

• 49J52
• 49J53
• 90C31

Acknowledgments

We would like to thank the two anonymous referees for taking their valuable time to read and give us their invaluable comments that help improve the content and the presentation of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Available at https://en.wikipedia.org/wiki/List of United States cities by population

Additional information

Funding

Research of this author was partly supported by the USA National Science Foundation (Directorate for Mathematical and Physical Sciences) [grant number DMS-1512846], [grant number DMS-1808978] and [grant number DMS-1716057], by the USA Air Force Office of Scientific Research [grant number #15RT04], and by Australian Research Council [grant number DP-190100555].

Notes on contributors

Anuj Bajaj

Anuj Bajaj is currently a PhD student in the Department of Mathematics at Wayne State University under the supervision of Dr Boris S. Mordukhovich with expected graduation date May 2021. He moved to Wayne State University in 2016 after completing his Master of Science (MSc) degree in Mathematics from The University of British Columbia. His research focusses on variational analysis and applied mathematics. Details about his educational background, publications and accomplishments can be found on his website https://www.sites.google.com/view/anujbajaj/

Boris S. Mordukhovich

Boris S. Mordukhovich received his PhD in Applied Mathematics from Belarus State University (Minsk, Belarus). He is currently a Distinguished University Professor in the Department of Mathematics at Wayne State University. His research focusses on Variational Analysis and Optimization, Systems Control and Operations Research, Nonlinear Dynamics, Applications to Economics, Engineering, Mechanics, and Behavioral Sciences. He is an Editorial Board member for many well-known research journals and is a Fellow of American Mathematical Society (AMS) and Society for Industrial and Applied Mathematics (SIAM). He has authored/coauthored various publications which can be found on his website https://borismordukhovich.com

Tuyen Tran

Tuyen Tran recently received her PhD in Mathematics from Portland State University under the supervision of Dr Nguyen Mau Nam. Her research focusses on Convex Optimization, Convex Analysis, DC programming, and Machine Learning. Starting Fall 2020 she will be joining Loyola University Chicago as an Assistant Professor.