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Canonical Duality Method for Solving Kantorovich Mass Transfer Problem

  • Xiaojun Lu
  • David Yang Gao
Chapter
Part of the Advances in Mechanics and Mathematics book series (AMMA, volume 37)

Abstract

This paper addresses analytical solution to the Kantorovich mass transfer problem . Through an ingenious approximation mechanism, the Kantorovich problem is first reformulated as a variational form, which is equivalent to a nonlinear differential equation with Dirichlet boundary. The existence and uniqueness of the solution can be demonstrated by applying the canonical duality theory. Then, using the canonical dual transformation, a perfect dual maximization problem is obtained, which leads to an analytical solution to the primal problem . Its global extremality for both primal and dual problems can be identified by a triality theory. In addition, numerical maximizers for the Kantorovich problem are provided under different circumstances. Finally, the theoretical results are verified by applications to Monge’s problem. Although the problem is addressed in one-dimensional space, the theory and method can be generalized to solve high-dimensional problems.

Notes

Acknowledgements

The main results in this paper were obtained during a research collaboration in the Federation University Australia in August, 2015. The first author wishes to thank Professor David Gao for his hospitality and financial support. This project is partially supported by US Air Force Office of Scientific Research (AFOSR FA9550-10-1-0487 and FA9550-17-1-0151). This project is also supported by Jiangsu Planned Projects for Postdoctoral Research Funds (1601157B), Shanghai University Start-up Grant for Shanghai 1000-Talent Program Scholars, National Natural Science Foundation of China (NSFC 61673104, 71673043, 71273048, 71473036, 11471072), the Scientific Research Foundation for the Returned Overseas Chinese Scholars, Fundamental Research Funds for the Central Universities (2014B15214, 2242017K40086), Open Research Fund Program of Jiangsu Key Laboratory of Engineering Mechanics, Southeast University (LEM16B06). In particular, the authors also express their deep gratitude to the referees for their careful reading and useful remarks.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.School of Mathematics and Jiangsu Key Laboratory of Engineering MechanicsSoutheast UniversityNanjingChina
  2. 2.Faculty of Science and TechnologyFederation University AustraliaBallaratAustralia

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