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Restricted Robinson Constraint Qualification and Optimality for Cardinality-Constrained Cone Programming

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Abstract

In this paper, optimality conditions are presented and analyzed for the cardinality-constrained cone programming arising from finance, statistical regression, signal processing, etc. By introducing a restricted form of (strict) Robinson constraint qualification, the first-order optimality conditions for the cardinality-constrained cone programming are established based upon the properties of the normal cone. After characterizing further the second-order tangent set to the cardinality-constrained system, the second-order optimality conditions are also presented under some mild conditions. These proposed optimality conditions, to some extent, enrich the optimization theory for noncontinuous and nonconvex programming problems.

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References

  1. Liu, J., Chen, J., Ye, J.: Large-scale sparse logistic regression. In: Proceedings of the 15th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 547–556. ACM (2009)

  2. Koh, K., Kim, S.J., Boyd, S.P.: An interior-point method for large-scale \(l_1\)-regularized logistic regression. J. Mach. Learn. Res. 8(8), 1519–1555 (2007)

    MathSciNet  MATH  Google Scholar 

  3. Bienstock, D.: Computational study of a family of mixed-integer quadratic programming problems. Math. Program. 74(2), 121–140 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  4. Gao, J., Li, D.: Optimal cardinality constrained portfolio selection. Oper. Res. 61(3), 745–761 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gao, J., Li, D.: A polynomial case of the cardinality-constrained quadratic optimization problem. J. Glob. Optim. 56(4), 1441–1455 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Li, D., Sun, X., Wang, J.: Optimal lot solution to cardinality constrained mean–variance formulation for portfolio selection. Math. Finance 16(1), 83–101 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Xu, F., Lu, Z., Xu, Z.: An efficient optimization approach for a cardinality-constrained index tracking problem. Optim. Method Softw. 31(2), 258–271 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Blumensath, T.: Compressed sensing with nonlinear observations and related nonlinear optimization problems. IEEE Trans. Inf. Theory 59(6), 3466–3474 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Candès, E.J., Tao, T.: Decoding by linear programming. IEEE Trans. Inf. Theory 51(12), 4203–4215 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Donoho, D.L.: Compressed sensing. IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chartrand, R.: Exact reconstruction of sparse signals via nonconvex minimization. IEEE Signal Proc. Lett. 14(10), 707–710 (2007)

    Article  Google Scholar 

  12. Chen, S., Donoho, D.: Basis pursuit. In: 1994 Conference Record of the Twenty-Eighth Asilomar Conference on Signals, Systems and Computers, 1994, vol. 1, pp. 41–44. IEEE (1994)

  13. Chen, X., Ge, D., Wang, Z., Ye, Y.: Complexity of unconstrained \(L_2-L_p\) minimization. Math. Program. 143(1–2), 371–383 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fan, J., Li, R.: Variable selection via nonconcave penalized likelihood and its oracle properties. J. Am. Stat. Assoc. 96(456), 1348–1360 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kong, L., Sun, J., Tao, J., Xiu, N.: Sparse recovery on Euclidean Jordan algebras. Linear Algebra Appl. 465, 65–87 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Zhang, C.H.: Nearly unbiased variable selection under minimax concave penalty. Ann. Stat. 38(2), 894–942 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Zhao, Y.B.: RSP-based analysis for sparsest and least \(l_1\)-norm solutions to underdetermined linear systems. IEEE Trans. Signal Proc. 61(22), 5777–5788 (2013)

    Article  Google Scholar 

  18. Bauschke, H.H., Luke, D.R., Phan, H.M., Wang, X.: Restricted normal cones and sparsity optimization with affine constraints. Found. Comput. Math. 14(1), 63–83 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Beck, A., Eldar, Y.C.: Sparsity constrained nonlinear optimization: optimality conditions and algorithms. SIAM J. Optim. 23(3), 1480–1509 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  20. Beck, A., Hallak, N.: On the minimization over sparse symmetric sets: projections, optimality conditions, and algorithms. Math. Oper. Res. 41(1), 196–223 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. Burdakov, O., Kanzow, C., Schwartz, A.: Mathematical programs with cardinality constraints: reformulation by complementarity-type constraints and a regularization method. SIAM J. Optim. 26(1), 397–425 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  22. Červinka, M., Kanzow, C., Schwartz, A.: Constraint qualifications and optimality conditions for optimization problems with cardinality constraints. Math. Program. 160, 353–377 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  23. Li, X., Song, W.: The first-order necessary conditions for sparsity constrained optimization. J. Oper. Res. Soc. China 3(4), 521–535 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lu, Z., Zhang, Y.: Sparse approximation via penalty decomposition methods. SIAM J. Optim. 23(4), 2448–2478 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lu, Z.: Optimization over sparse symmetric sets via a nonmonotone projected gradient method. arXiv:1509.08581v3 (2015)

  26. Pan, L.L., Xiu, N.H., Zhou, S.L.: On solutions of sparsity constrained optimization. J. Oper. Res. Soc. China 3(4), 421–439 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  27. Bertsimas, D., Shioda, R.: Algorithm for cardinality-constrained quadratic optimization. Comput. Optim. Appl. 43(1), 1–22 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zou, H., Hastie, T., Tibshirani, R.: Sparse principal component analysis. J. Comput. Graph. Stat. 15(2), 265–286 (2006)

    Article  MathSciNet  Google Scholar 

  29. d’Aspremont, A., El Ghaoui, L., Jordan, M.I., Lanckriet, G.R.: A direct formulation for sparse PCA using semidefinite programming. SIAM Rev. 49(3), 434–448 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  30. Fukuda, M., Kojima, M., Murota, K., Nakata, K.: Exploiting sparsity in semidefinite programming via matrix completion I: general framework. SIAM J. Optim. 11(3), 647–674 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  31. Heiler, M., Schnörr, C.: Learning sparse representations by non-negative matrix factorization and sequential cone programming. J. Mach. Learn. Res. 7, 1385–1407 (2006)

    MathSciNet  MATH  Google Scholar 

  32. Rockafellar, R.T., Wets, R.J.: Variational Analysis. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  33. Clarke, F.H.: Optimization and Nonsmooth Analysis. SIAM, Philadelphia, PA (1990)

    Book  MATH  Google Scholar 

  34. Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, Berlin (2000)

    Book  MATH  Google Scholar 

  35. Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton, NJ (2015)

    MATH  Google Scholar 

  36. Flegel, M.L., Kanzow, C., Outrata, J.V.: Optimality conditions for disjunctive programs with application to mathematical programs with equilibrium constraints. Set-Valued Anal. 15(2), 139–162 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  37. Moldovan, A., Pellegrini, L.: On regularity for constrained extremum problems. Part 1: sufficient optimality conditions. J. Optim. Theory Appl. 142(1), 147–163 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  38. Moldovan, A., Pellegrini, L.: On regularity for constrained extremum problems. Part 2: necessary optimality conditions. J. Optim. Theory Appl. 142(1), 165–183 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The work was supported in part by the National Natural Science Foundation of China (Grants 11431002, 11771038, 11771255) and Shandong Province Natural Science Foundation (Grant ZR2016AM07). The authors thank the editor and referees for their useful suggestions and comments that improved the quality of the paper.

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Authors and Affiliations

  1. Department of Applied Mathematics, Beijing Jiaotong University, Beijing, 100044, People’s Republic of China

    Lili Pan & Naihua Xiu

  2. Department of Mathematics, Shandong University of Technology, Zibo, 255049, People’s Republic of China

    Lili Pan

  3. State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing, 100044, People’s Republic of China

    Ziyan Luo

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  1. Lili Pan
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  2. Ziyan Luo
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  3. Naihua Xiu
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Correspondence to Lili Pan.

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Pan, L., Luo, Z. & Xiu, N. Restricted Robinson Constraint Qualification and Optimality for Cardinality-Constrained Cone Programming. J Optim Theory Appl 175, 104–118 (2017). https://doi.org/10.1007/s10957-017-1166-4

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  • DOI: https://doi.org/10.1007/s10957-017-1166-4

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