A new algorithm for concave quadratic programming

Abstract

The main outcomes of the paper are divided into two parts. First, we present a new dual for quadratic programs, in which, the dual variables are affine functions, and we prove strong duality. Since the new dual is intractable, we consider a modified version by restricting the feasible set. This leads to a new bound for quadratic programs. We demonstrate that the dual of the bound is a semi-definite relaxation of quadratic programs. In addition, we probe the relationship between this bound and the well-known bounds in the literature. In the second part, thanks to the new bound, we propose a branch and cut algorithm for concave quadratic programs. We establish that the algorithm enjoys global convergence. The effectiveness of the method is illustrated for numerical problem instances.

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Acknowledgements

I am very grateful to two anonymous referees for their valuable comments and suggestions which help to improve the paper considerably. The author would like to thank associate editor for the very thoughtful comments provided on the first version of this manuscript. This research was in part supported by a grant from the Iran National Science Foundation (No. 96010653, Principal investigator: Dr. Majid Soleimani-damaneh).

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Correspondence to Moslem Zamani.

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Zamani, M. A new algorithm for concave quadratic programming. J Glob Optim 75, 655–681 (2019). https://doi.org/10.1007/s10898-019-00787-w

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Keywords

  • Non-convex quadratic programming
  • Duality
  • Semi-definite relaxation
  • Bound
  • Branch and cut method
  • Concave quadratic programming