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On the coupled continuous knapsack problems: projection onto the volume constrained Gibbs N-simplex

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Abstract

Coupled continuous quadratic knapsack problems (CCK) are introduced in the present study. The solution of a CCK problem is equivalent to the projection of an arbitrary point onto the volume constrained Gibbs N-simplex, which has a wide range of applications in computational science and engineering. Three algorithms have been developed in the present study to solve large scale CCK problems. According to the numerical experiments of this study, the computational costs of presented algorithms scale linearly with the problem size, when it is sufficiently large. Moreover, they show competitive or even superior computational performance compared to the advanced QP solvers. The ease of implementation and linear scaling of memory usage with the problem size are the other benefits of the presented algorithms.

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Notes

  1. For a comprehensive account on bibliography and software corresponding to solution of QP problems visit: http://www.numerical.rl.ac.uk/people/nimg/qp/qp.html.

  2. http://cvxr.com/cvx/.

  3. http://sedumi.ie.lehigh.edu/.

  4. http://www.math.nus.edu.sg/~mattohkc/sdpt3.html.

  5. http://gurobi.com/.

  6. http://mosek.com/.

  7. More precisely, maximum attainable accuracy by the solver.

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Acknowledgments

We would like to thanks Ernesto G. Birgin for the suggestion of alternating projection algorithm for the solution of problem studied in this work. Helpful comments by two anonymous reviewers which improve the presentation of our paper are greatly acknowledged.

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  1. Department of Materials Science and Engineering, Sharif University of Technology, P.O. Box 11365-9466, Tehran, Iran

    Rouhollah Tavakoli

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  1. Rouhollah Tavakoli
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Tavakoli, R. On the coupled continuous knapsack problems: projection onto the volume constrained Gibbs N-simplex. Optim Lett 10, 137–158 (2016). https://doi.org/10.1007/s11590-015-0866-7

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