Tilt Stability for Quadratic Programs with One or Two Quadratic Inequality Constraints

Abstract

This paper examines tilt stability for quadratic programs with one or two quadratic inequality constraints. Exploiting specific features of these problems and using some known results on tilt stability in nonlinear programming, we establish quite simple characterizations of tilt-stable local minimizers for quadratic programs with one quadratic inequality constraint under metric subregularity constraint qualification. By the same way, we also derive various tilt stability conditions for quadratic programs with two quadratic inequality constraints and satisfying certain suitable assumptions. Especially, the obtained results show that some tilt stability conditions only known to be sufficient in nonlinear programming become the necessary ones when the considered problems are quadratic programs with one or two quadratic inequality constraints.

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References

  1. 1.

    Ai, W., Zhang, S.: Strong duality for the CDT subproblem: a necessary and sufficient condition. SIAM J. Optim. 19, 1735–1756 (2008)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Bomze, I. M., Overton, M. L.: Narrowing the difficulty gap for the Celis-Dennis-Tapia problem. Math. Program. 151, 459–476 (2015)

    MathSciNet  Article  Google Scholar 

  3. 3.

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

    Google Scholar 

  4. 4.

    Beck, A., Vaisbourd, Y.: Globally Solving the trust region subproblem using simple first-order methods. SIAM J. Optim. 28, 1951–1967 (2018)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Bienstock, D.: A note on polynomial solvability of the CDT problem. SIAM J. Optim. 26, 488–498 (2016)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Celis, M. R., Dennis, J. E., Tapia, R. A.: A trust region algorithm for nonlinear equality constrained optimization. In: Optimization Numerical, Boggs, P. T., Byrd, R. H., Schnabel, R. B. (eds.) , pp 71–82. SIAM, Philadelphia (1985)

  7. 7.

    Chieu, N. H., Hien, L. V.: Computation of graphical derivative for a class of normal cone mappings under a very weak condition. SIAM J. Optim. 27, 190–204 (2017)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Chieu, N. H., Hien, L. V., Nghia, T. T. A.: Characterization of tilt stability via subgradient graphical derivative with applications to nonlinear programming. SIAM J. Optim. 28, 2246–2273 (2018)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Conn, A. R., Gould, N. I. M., Toint, P.H.L.: Trust-Region methods MPS-SIAM series on optimization, vol. 01. SIAM, Philadelphia (2000)

    Google Scholar 

  10. 10.

    Consolini, L., Locatelli, M.: On the complexity of quadratic programming with two quadratic constraints. Math. Program. 164, 91–128 (2017)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Drusvyatskiy, D., Lewis, A. S.: Tilt stability, uniform quadratic growth, and strong metric regularity of the subdifferential. SIAM J. Optim. 23, 256–267 (2013)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Eberhard, A. C., Wenczel, R.: A study of tilt-stable optimality and suffcient conditions. Nonlinear Anal. 75, 1260–1281 (2012)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Fortin, C., Wolkowicz, H.: The trust region subproblem and semidefinite programming. Optim. Methods Software 19, 41–67 (2004)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Gfrerer, H., Mordukhovich, B. S.: Complete characterizations of tilt stability in nonlinear programming under weakest qualification conditions. SIAM J. Optim. 25, 2081–2119 (2015)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Gould, N. I. M., Lucidi, S., Roma, M., Toint, P. L.: Solving the trust-region subproblem using the Lanczos method. SIAM J. Optim. 9, 504–525 (1999)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Gould, N. I. M., Toint, P.H.L.A: Quadratic programming bibliography. Numerical Analysis Group Internal Report 2000-1, Rutherford Appleton Laboratory, Chilton, Oxfordshire UK (2000)

  17. 17.

    Lee, G. M., Tam, N. N., Yen, N. D.: Quadratic Programming and Affine Variational Inequalities: a Qualitative Study. Springer-Verlag, New York (2005)

    Google Scholar 

  18. 18.

    Lee, G. M., Tam, N. N., Yen, N. D.: Stability of linear-quadratic minimization over Euclidean balls. SIAM J. Optim. 22, 936–952 (2012)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Levy, A. B., Poliquin, R. A., Rockafellar, R. T.: Stability of locally optimal solutions. SIAM J. Optim. 10, 580–604 (2000)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Lewis, A. S., Zhang, S.: Partial smoothness, tilt stability, and generalized Hessians. SIAM J. Optim. 23, 74–94 (2013)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Martínez, J. M.: Local minimizers of quadratic functions on Euclidean balls and spheres. SIAM J. Optim. 4, 159–176 (1994)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Mordukhovich, B.S.: Maximum principle in problems of time optimal control with nonsmooth constraints. J. Appl. Math. Mech. 40, 960–969 (1976)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Mordukhovich, B. S.: Variational Analysis and Generalized Differentiation I: Basic Theory. Springer, Berlin (2006)

    Google Scholar 

  24. 24.

    Mordukhovich, B. S., Nghia, T. T. A.: Second-order characterizations of tilt stability with applications to nonlinear programming. Math. Program. 149, 83–104 (2015)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Mordukhovich, B. S., Outrata, J. V.: Tilt stability in nonlinear programming under Mangasarian-Fromovitz constraint qualification. Kybernetika 49, 446–464 (2012)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Mordukhovich, B. S., Rockafellar, R. T.: Second-order subdifferential calculus with applications to tilt stability in optimization. SIAM J. Optim. 22, 953–986 (2012)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Nocedal, J., Wright, S. J.: Numerical Optimization. Springer, New York (2006)

    Google Scholar 

  28. 28.

    Nghi, T. V.: Coderivatives related to parametric extended trust region subproblem and their applications. Taiwanese J. Math. 22, 485–511 (2018)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Pham Dinh, T., Le Thi, H. A.: A DC optimization algorithm for solving the trust-region subproblem. SIAM J. Optim. 8, 476–505 (1998)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Peng, J. M., Yuan, Y.: Optimality conditions for the minimization of a quadratic with two quadratic constraints. SIAM J. Optim. 7, 579–594 (1997)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Poliquin, R. A., Rockafellar, R. T.: Tilt stability of a local minimum. SIAM J. Optim. 8, 287–299 (1998)

    MathSciNet  Article  Google Scholar 

  32. 32.

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

    Google Scholar 

  33. 33.

    Rojas, M., Santos, S. A., Sorensen, D. C.: A new matrix-free algorithm for the large-scale trust-region subproblem. SIAM J. Optim. 11, 611–646 (2000)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Sakaue, S., Nakatsukasa, Y., Takeda, A., Iwata, S.: Solving generalized CDT problems via two-parameter eigenvalues. SIAM J. Optim. 26, 1669–1694 (2016)

    MathSciNet  Article  Google Scholar 

  35. 35.

    Salahi, M., Taati, A., Wolkowicz, H.: Local nonglobal minima for solving large-scale extended trust-region subproblems. Comput. Optim. Appl. 66, 223–244 (2017)

    MathSciNet  Article  Google Scholar 

  36. 36.

    Tuy, H.: Convex Analysis and Global Optimization (second edition). Springer International Publishing AG (2016)

  37. 37.

    Tuy, H., Tuan, H. D.: Generalized S-Lemma and strong duality in nonconvex quadratic programming. J. Global Optim. 56, 1045–1072 (2013)

    MathSciNet  Article  Google Scholar 

  38. 38.

    Ye, Y.: On affine scaling algorithms for nonconvex quadratic programming. Math. Program. 56, 285–300 (1992)

    MathSciNet  Article  Google Scholar 

  39. 39.

    Ye, Y.: A new complexity result on minimization of a quadratic function with a sphere constraint. In: Recent Advances in Global Optimization, 19–31. Princeton University Press (1992)

  40. 40.

    Ye, Y., Zhang, S.: New results on quadratic minimization. SIAM J. Optim. 14, 245–267 (2003)

    MathSciNet  Article  Google Scholar 

  41. 41.

    Yuan, Y.: On a subproblem of trust region algorithms for constrained optimization. Math. Program. 47, 53–63 (1990)

    MathSciNet  Article  Google Scholar 

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Acknowledgments

The authors would like to thank the referee and the Handling Editor for their valuable comments and kind suggestions.

Funding

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2017.325.

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Correspondence to Nguyen Huy Chieu.

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Dedicated to Professor Hoang Tuy on the occasion of his 90th birthday

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Chieu, N.H., Van Hien, L. & Trang, N.T.Q. Tilt Stability for Quadratic Programs with One or Two Quadratic Inequality Constraints. Acta Math Vietnam 45, 477–499 (2020). https://doi.org/10.1007/s40306-020-00372-4

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Keywords

  • Tilt stability
  • Strong second-order sufficient condition
  • Metric subregularity constraint qualification
  • Quadratic program
  • Quadratic inequality constraint

Mathematics Subject Classification (2010)

  • 49J53
  • 90C31
  • 90C46