Skip to main content
SpringerLink
Log in

Sufficient Conditions for Metric Subregularity of Constraint Systems with Applications to Disjunctive and Ortho-Disjunctive Programs

  • Published:
Set-Valued and Variational Analysis Aims and scope Submit manuscript

Abstract

This paper is devoted to the study of the metric subregularity constraint qualification for general optimization problems, with the emphasis on the nonconvex setting. We elaborate on notions of directional pseudo- and quasi-normality, recently introduced by Bai et al., which combine the standard approach via pseudo- and quasi-normality with modern tools of directional variational analysis. We focus on applications to disjunctive programs, where (directional) pseudo-normality is characterized via an extremal condition. This, in turn, yields efficient tools to verify pseudo-normality and the metric subregularity constraint qualification, which include, but are not limited to, Robinson’s result on polyhedral multifunctions and Gfrerer’s second-order sufficient condition for metric subregularity. Finally, we refine our study by defining the new class of ortho-disjunctive programs which comprises prominent optimization problems such as mathematical programs with complementarity, vanishing or switching constraints.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Achtziger, W., Kanzow, C.: Mathematical programs with vanishing constraints: Optimality conditions and constraint qualifications. Math. Program. 114, 69–99 (2008)

  2. Adam, L., Branda, M.: Nonlinear chance constrained problems: optimality conditions, regularization and solvers. J. Optim. Theory Appl. 170(2), 419–436 (2016)

  3. Bai, K., Ye, J.J., Zhang, J.: Directional quasi-/pseudo-normality as sufficient conditions for metric subregularity. SIAM J. Optim., 29(4), 2625–2649 (2019)

  4. Benko, M., Gfrerer, H.: On estimating the regular normal cone to constraint systems and stationary conditions. Optimization 66, 61–92 (2017)

  5. Benko, M., Gfrerer, H.: New verifiable stationary concepts for a class of mathematical programs with disjunctive constraints. Optimization 67, 1–23 (2018)

  6. Benko, M.: Numerical methods for mathematical programs with disjunctive constraints. PhD-thesis, Univ. Linz (2016)

  7. Benko, M., Gfrerer, H., Outrata, J.V.: Calculus for directional limiting normal cones and subdifferentials. Set-Valued Var. Anal. 27(3), 713–745 (2019)

  8. Benko, M., Gfrerer, H., Mordukhovich, B.S.: Characterizations of tilt-stable minimizers in second-order cone programming. SIAM J. Optim. 29(4), 3100–3130 (2019)

  9. Benko, M., Gfrerer, H., Outrata, J.V., Stability analysis for parameterized variational systems with implicit constraints. Set-Valued Var. Anal. 28, 167–193 (2019)

  10. Bertsekas, D., Ozdaglar, A.E.: Pseudonormality and a Lagrange multiplier theory for constrained optimization. J. Optim. Theory Appl. 114, 287–343 (2002)

  11. Bigi, G.: On sufficient second order optimality conditions in multiobjective optimization. Math. Meth. Oper. Res. 63, 77–85 (2006)

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

  13. Burke, J.V.: Calmness and exact penalization. SIAM J. Control and Optim. 29, 493–497 (1991)

  14. Burke, J.V.: An exact penalization viewpoint of constrained optimization. SIAM J. Control Optim. 29, 968–998 (1991)

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

  16. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

  17. Dempe, S.: Foundations of Bilevel Programming. Nonconvex Optimization and Its Applications, vol. 61. Kluwer Academic Publishers, Dordrecht (2002)

  18. Dontchev, A.L., Rockafellar, R.T.: Regularity and conditioning of solution mappings in variatonal analysis, Set-Valued Anal. 12, 79–109 (2004)

  19. Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings. Springer, Heidelberg (2014)

  20. Ehrgott, M.: Multicriteria Optimization. Springer, Berlin (2005)

  21. Fabian, M., Henrion, R., Kruger, A.Y., Outrata, J.V.: Error bounds: necessary and sufficient conditions. Set-Valued Var. Anal. 18, 121–149 (2010)

  22. 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, 139–162 (2007)

  23. Gfrerer, H.: On directional metric regularity, subregularity and optimality conditions for nonsmooth mathematical programs. Set-Valued Var. Anal. 21, 151–176 (2013)

  24. Gfrerer, H.: On directional metric subregularity and second-order optimality conditions for a class of nonsmooth mathematical programs. SIAM J. Optim. 23, 632–665 (2013)

  25. Gfrerer, H.: Optimality conditions for disjunctive programs based on generalized differentiation with application to mathematical programs with equilibrium constraints. SIAM J. Optim. 24, 898–931 (2014)

  26. Gfrerer, H.: On metric pseudo-(sub)regularity of multifunctions and optimality conditions for degenerated mathematical programs. Set-Valued Var. Anal. 22, 79–115 (2014)

  27. Gfrerer, H., Klatte, D.: Lipschitz and Hölder stability of optimization problems and generalized equations. Math. Program. 158, 35–75 (2016)

  28. Gfrerer, H., Outrata, J.V.: On Lipschitzian properties of implicit multifunctions. SIAM J. Optim. 26, 2160–2189 (2016)

  29. Gfrerer, H., Mordukhovich, B.S.: Second-order variational analysis of parametric constraint and variational systems. SIAM J. Optim. 29, 423–453 (2019)

  30. Guo, L., Ye, J.J., Zhang, J.: Mathematical programs with geometric constraints in banach spaces: enhanced optimality, exact penalty, and sensitivity. SIAM J. Optim. 23, 2295–2319 (2013)

  31. Henrion, R., Jourani, A., Outrata, J.V.: On the calmness of a class of multifunctions. SIAM J. Optim. 13, 603–618 (2002)

  32. Henrion, R., Outrata, J.V.: Calmness of constraint systems with applications. Math. Program. 104 437–464 (2005)

  33. Henrion, R., Outrata, J.V.: On calculating the normal cone to a finite union of convex polyhedra. Optimization 57(1), 57–78 (2008)

  34. Hiriart-Urrruty, J.-B., Lemaréchal, C.: Fundamentals of Convex Analysis. Grundlehren Text Editions. Springer, Berlin (2001)

  35. Hoheisel, T.: Mathematical programs with vanishing constraints. PhD-thesis, Julius–Maximilians–Universität Würzburg (2009)

  36. Hu, Q., Zhang, H., Chen, Y., Tang, M.: An Improved Exact Penalty Result for Mathematical Programs with Vanishing Constraints. J. Adv. Appl. Math. 3, 43–49 (2018)

  37. Ioffe, A.D.: Variational Analysis of Regular Mappings. Springer Monographs in Mathematics. Springer, Cham (2017)

  38. Ioffe, A.D., Outrata, J.V.: On metric and calmness qualification conditions in subdifferential calculus. Set-Valued Anal. 16, 199–227 (2018)

  39. Jahn, J.: Vector Optimization: Theory, Applications, and Extensions. Springer, Berlin (2011)

  40. Kanzow, C., Schwartz, A.: Mathematical Programs with Equilibrium Constraints: Enhanced Fritz John-conditions, New Constraint Qualifications, and Improved Exact Penalty Results. SIAM J. Optim. 20, 2730–2753 (2010)

  41. Klatte, D., Kummer, B.: Constrained minima and Lipschitzian penalties in metric spaces. SIAM J. Optim. 13, 619–633 (2002)

  42. Klatte, D., Kummer, B.: Nonsmooth Equations in Optimization: Regularity, Calculus, Methods and Applications. Nonconvex Optimization and Its Applications, vol. 60, Kluwer, Dordrecht/Boston/London (2002)

  43. Kruger, A.Y.: Error bounds and metric subregularity. Optimization 64, 49–79 (2015)

  44. Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996)

  45. Mehlitz, P.: On the linear independence constraint qualification in disjunctive programming. Optimization, pp. 37. https://doi.org/10.1080/02331934.2019.1679811 (2019)

  46. Mehlitz, P.: Stationarity conditions and constraint qualifications for mathematical programs with switching constraints. Math. Program. 181(1), 149–186 (2020)

  47. Mordukhovich, B.S.: Variational analysis and applications. Springer, Cham (2018)

  48. Outrata, J.V., Kočvara, M., Zowe, J.: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Nonconvex Optimization and its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands (1998)

  49. Penot, J.P.: Error bounds, calmness and their applications in nonsmooth analysis. In: Nonlinear Analysis and Optimization II: Optimization, A. Leizarowitz, B.S. Mordukhovich, I. Shafrir, Zaslavski, A.J. eds., vol. 514 of Contemp. Math., AMS, Providence, pp. 225–248 (2010)

  50. Robinson, S.M.: Some continuity properties of polyhedral multifunctions. Math. Program. Studies 14, 206–214 (1981)

  51. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. A Series of Comprehensive Studies in Mathematics vol. 317, Springer, Berlin (1998)

  52. Scholtes, S.: Nonconvex structures in nonlinear programming. Oper. Res. 52, 368-383 (2004)

  53. Stein, O.: Bi-level Strategies in Semi-Infinite Programming. Nonconvex Optimization and Its Applications, vol. 71, Kluwer, Boston (2003)

  54. Wu, Z., Ye, J.J.: First-order and second-order conditions for error bounds. SIAM J. Optim. 14(3), 621–645 (2003)

  55. Ye, J.J., Zhang, J.: Enhanced Karush-Kuhn-Tucker conditions for mathematical programs with equilibrium constraints. J. Optim. Theory Appl. 164, 777–794 (2014)

Download references

Acknowledgments

The authors also thank two anonymous referees for their comments which helped improve the presentation of the material.

Funding

The research of the first author was supported by the Austrian Science Fund (FWF) under grant P29190-N32. The work on the revised version was supported by the FWF grant P32832-N. The research of the second author was supported by the Grant Agency of the Czech Republic (Grant No. 18-04145S). Part of this work was done while the second author was visiting McGill University, partially supported by H2020-MSCA-RISE project GEMCLIME-2020 under GA No. 681228. The research of the third author was supported by an NSERC discovery grant.

Author information

Authors and Affiliations

  1. Institute of Computational Mathematics, Johannes Kepler University Linz, A-4040, Linz, Austria

    Matúš Benko

  2. Faculty of Mathematics, University of Vienna, 1090, Vienna, Austria

    Matúš Benko

  3. Institute of Economic Studies, Faculty of Social Sciences, Charles University, Opletalova 26, 110 00, Prague 1, Czech Republic

    Michal Červinka

  4. Institute of Information Theory and Automation, Czech Academy of Sciences, Pod Vodarenskou vezi 4, 180 00, Prague 8, Czech Republic

    Michal Červinka

  5. Institute of Mathematics and Statistics, McGill University, 805 Sherbrooke St West, Room 1114 Montréal, Québec, H3A 0B9, Canada

    Tim Hoheisel

Authors
  1. Matúš Benko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michal Červinka
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tim Hoheisel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tim Hoheisel.

Additional information

The authors would like to dedicate this paper to Helmut Gfrerer in honor of his 60th birthday.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Benko, M., Červinka, M. & Hoheisel, T. Sufficient Conditions for Metric Subregularity of Constraint Systems with Applications to Disjunctive and Ortho-Disjunctive Programs. Set-Valued Var. Anal 30, 143–177 (2022). https://doi.org/10.1007/s11228-020-00569-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11228-020-00569-7

Keywords

Mathematics Subject Classification (2010)

Search

Navigation