Global Optimization from Concave Minimization to Concave Mixed Variational Inequality

Abstract

We use techniques from global optimization to develop an algorithm for finding a global solution of nonconvex mixed variational inequality problems involving separable DC cost functions. In contrast to the convex mixed variational inequality, in these problems, a local solution may not be a global one. The proposed algorithm uses the convex envelope of the separable cost function over boxes to approximate a DC cost problem with a convex cost one that can be solved by available methods. To obtain better approximate solutions, the algorithm uses an adaptive rectangular bisection which is performed only in the space of concave variables. The algorithm is applied to solve the Nash-Cournot and Bertrand equilibrium models with logarithm and quadratic concave costs. Computational results on a lot number of randomly generated data show that the proposed algorithm is efficient for these models, when the number of the concave cost functions is moderate, while the size of the model may be much larger.

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Acknowledgements

We would like to thank the editor and referees for their valuable comments, suggestions, and remarks that helped us very much to improve the quality of the paper.

Funding

The first author of this paper appreciated the support from the NAFOSTED, under grant 101.01-2017.315.

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Correspondence to Le Dung Muu.

Additional information

This paper is dedicated to Professor Hoang Tuy on the occasion of his 90th birthday

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Muu, L.D., Van Quy, N. Global Optimization from Concave Minimization to Concave Mixed Variational Inequality. Acta Math Vietnam 45, 449–462 (2020). https://doi.org/10.1007/s40306-020-00363-5

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Keywords

  • DC optimization
  • Nonconvex mixed variational inequality
  • Nash-Cournot oligopolistic model
  • Concave cost
  • Global solution
  • Gap function
  • Convex envelope

Mathematics Subject Classification (2010)

  • 47J25
  • 47N10
  • 90C25