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Random projections for quadratic programs

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Abstract

Random projections map a set of points in a high dimensional space to a lower dimensional one while approximately preserving all pairwise Euclidean distances. Although random projections are usually applied to numerical data, we show in this paper that they can be successfully applied to quadratic programming formulations over a set of linear inequality constraints. Instead of solving the higher-dimensional original problem, we solve the projected problem more efficiently. This yields a feasible solution of the original problem. We prove lower and upper bounds of this feasible solution w.r.t. the optimal objective function value of the original problem. We then discuss some computational results on randomly generated instances, as well as a variant of Markowitz’ portfolio problem. It turns out that our method can find good feasible solutions of very large instances.

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References

  1. Achlioptas, D.: Database-friendly random projections: Johnson–Lindenstrauss with binary coins. J. Comput. Syst. Sci. 66, 671–687 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ailon, N., Chazelle, B.: Approximate nearest neighbors and fast Johnson–Lindenstrauss lemma. In: Proceedings of the Symposium on the Theory Of Computing, Volume ’06 of STOC. ACM, Seattle (2006)

  3. Boutsidis, C., Zouzias, A., Drineas, P.: Random projections for \(k\)-means clustering. In: Advances in Neural Information Processing Systems, NIPS. NIPS Foundation, La Jolla, pp. 298–306 (2010)

  4. Dasgupta, S.: Experiments with random projection. In: Proceedings of the 16th Conference on Uncertainty in Artificial Intelligence, pp. 143–151. Morgan Kaufman, San Francisco (2000)

  5. Dorn, W.: Duality in quadratic programming. Q. Appl. Math. 18(2), 155–162 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  6. Gould, N., Toint, P.: A quadratic programming bibliography. In: Technical Report 2000–2001. RAL Numerical Analysis Group (2001)

  7. IBM: ILOG CPLEX 12.8 User’s Manual. IBM (2017)

  8. Indyk, P., Motwani, R.: Approximate nearest neighbors: towards removing the curse of dimensionality. In: Proceedings of the Symposium on the Theory Of Computing, Volume 30 of STOC, pp. 604–613. ACM, New York (1998)

  9. Indyk, P., Naor, A.: Nearest neighbor preserving embeddings. ACM Trans. Algorithms 3(3), 31 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Johnson, W., Lindenstrauss, J.: Extensions of Lipschitz mappings into a Hilbert space. In: Hedlund, G. (ed.) Conference in Modern Analysis and Probability. Contemporary Mathematics, vol. 26, pp. 189–206. AMS, Providence, RI (1984)

    Chapter  Google Scholar 

  11. Kane, D., Nelson, J.: Sparser Johnson–Lindenstrauss transforms. J. ACM 61(1), 4 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Liberti, L., Vu, K.: Barvinok’s naive algorithm in distance geometry. Oper. Res. Lett. 46, 476–481 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  13. Markowitz, H.: Portfolio selection. J. Finance 7(1), 77–91 (1952)

    Google Scholar 

  14. May, J., Smith, R.: Random polytopes: their definition, generation and aggregate properties. Math. Program. 24, 39–54 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  15. Paul, S., Boutsidis, C., Magdon-Ismail, M., Drineas, P.: Random projections for linear support vector machines. ACM Trans Knowl Discov Data 8(4), 22:1–22:25 (2014)

    Article  Google Scholar 

  16. Pilanci, M., Wainwright, M.: Randomized sketches of convex programs with sharp guarantees. In: International Symposium on Information Theory (ISIT), pp. 921–925. IEEE, Piscataway (2014)

  17. Pilanci, M., Wainwright, M.: Newton sketch: a linear time optimization algorithm with linear-quadratic convergence. SIAM J. Optim. 27(1), 205–245 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  18. Shim, J.: A survey of quadratic programming applications to business and economics. Int. J. Syst. Sci. 14(1), 105–115 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  19. Vavasis, S.: Quadratic programming is in NP. Inf. Process. Lett. 36, 73–77 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  20. Vempala, S.: The random projection method. In: Number 65 in DIMACS Series in Discrete Mathematics and Theoretical Computer Science. AMS, Providence, RI (2004)

  21. Venkatasubramanian, S., Wang, Q.: The Johnson–Lindenstrauss transform: an empirical study. In: Algorithm Engineering and Experiments, Volume 13 of ALENEX, pp. 164–173. SIAM, Providence, RI (2011)

  22. Vershynin, R.: High-dimensional Probability. CUP, Cambridge (2018)

    Book  MATH  Google Scholar 

  23. Vu, K., Poirion, P.-L., D’Ambrosio, C., Liberti, L.: Random projections for quadratic programs over a Euclidean ball. In: Lodi, A., et al. (eds.) Integer Programming and Combinatorial Optimization (IPCO). LNCS, vol. 11480, pp. 442–452. Springer, New York (2019)

    Chapter  Google Scholar 

  24. Vu, K., Poirion, P.-L., Liberti, L.: Random projections for linear programming. Math. Oper. Res. 43(4), 1051–1071 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  25. Vu, K., Poirion, P.-L., Liberti, L.: Gaussian random projections for Euclidean membership problems. Discrete Appl. Math. 253, 93–102 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  26. Woodruff, D.: Sketching as a tool for linear algebra. Found. Trends Theor. Comput. Sci. 10(1–2), 1–157 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  27. Yang, J., Meng, X., Mahoney, M.: Quantile regression for large-scale applications. SIAM J. Sci. Comput. 36(5), S78–S110 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zhang, L., Mahdavi, M., Jin, R., Yang, T., Zhu, S.: Recovering the optimal solution by dual random projection. In: Shalev-Shwartz S., Steinwart I. (eds) Conference on Learning Theory (COLT), Volume 30 of Proceedings of Machine Learning Research, pp. 135–157. \(\langle \)jmlr.org\(\rangle \) (2013)

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Acknowledgements

We are grateful to two anonymous referees for many excellent suggestions and insights.

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Authors and Affiliations

  1. CNRS LIX École Polytechnique, Institut Polytechnique de Paris, Palaiseau, France

    Claudia D’Ambrosio & Leo Liberti

  2. RIKEN Center for Advanced Intelligence Project, Tokyo, Japan

    Pierre-Louis Poirion

  3. Department of Mathematics, FPT University, Hoa Lac Hi-Tech Park, Hanoi, Vietnam

    Ky Vu

Authors
  1. Claudia D’Ambrosio
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  2. Leo Liberti
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  3. Pierre-Louis Poirion
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  4. Ky Vu
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Correspondence to Leo Liberti.

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This paper has received Funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant Agreement n. 764759 “MINOA”

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D’Ambrosio, C., Liberti, L., Poirion, PL. et al. Random projections for quadratic programs. Math. Program. 183, 619–647 (2020). https://doi.org/10.1007/s10107-020-01517-x

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