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R-Regularity of Set-Valued Mappings Under the Relaxed Constant Positive Linear Dependence Constraint Qualification with Applications to Parametric and Bilevel Optimization

Abstract

The presence of Lipschitzian properties for solution mappings associated with nonlinear parametric optimization problems is desirable in the context of, e.g., stability analysis or bilevel optimization. An example of such a Lipschitzian property for set-valued mappings, whose graph is the solution set of a system of nonlinear inequalities and equations, is R-regularity. Based on the so-called relaxed constant positive linear dependence constraint qualification, we provide a criterion ensuring the presence of the R-regularity property. In this regard, our analysis generalizes earlier results of that type which exploited the stronger Mangasarian–Fromovitz or constant rank constraint qualification. Afterwards, we apply our findings in order to derive new sufficient conditions which guarantee the presence of R-regularity for solution mappings in parametric optimization. Finally, our results are used to derive an existence criterion for solutions in pessimistic bilevel optimization and a sufficient condition for the presence of the so-called partial calmness property in optimistic bilevel optimization.

References

  1. Andreani, R., Haeser, G., Schuverdt, M.L., Silva, P.J.S.: A relaxed constant positive linear dependence constraint qualification and applications. Math. Program. 135(1), 255–273 (2012). https://doi.org/10.1007/s10107-011-0456-0

    MathSciNet  MATH  Google Scholar 

  2. Andreani, R., Martinez, J.M., Schuverdt, M.L.: On the relation between constant positive linear dependence condition and quasinormality constraint qualification. J. Optim. Theory Appl. 125(2), 473–483 (2005). https://doi.org/10.1007/s10957-004-1861-9

    MathSciNet  MATH  Google Scholar 

  3. Bai, K., Ye, J.J.: Directional necessary optimality conditions for bilevel programs. arXiv:https://arxiv.org/abs/2004.01783,1–31 (2020)

  4. Bank, B., Guddat, J., Klatte, D., Kummer, B., Tammer, K.: Nonlinear Parametric Optimization. Basel, Birkhäuser (1983)

    MATH  Google Scholar 

  5. Bard, J.F.: Practical Bilevel Optimization: Algorithms and Applications. Kluwer Academic, Dordrecht (1998)

    MATH  Google Scholar 

  6. Bednarczuk, E.M., Minchenko, L.I., Rutkowski, K.E.: On Lipschitz-like continuity of a class of set-valued mappings. Optimization 69(12), 2535–2549 (2020). https://doi.org/10.1080/02331934.2019.1696339

    MathSciNet  MATH  Google Scholar 

  7. Bertsekas, D.P.: Nonlinear Programming. Athena Scientific, Belmot (1999)

    MATH  Google Scholar 

  8. Borwein, J.M.: Stability and regular points of inequality systems. J. Optim. Theory Appl. 48(1), 9–52 (1986). https://doi.org/10.5555/3182697.3183275

    MathSciNet  MATH  Google Scholar 

  9. Bosch, P., Jourani, A., Henrion, R.: Sufficient conditions for error bounds and applications. Appl. Math. Optim. 50(2), 161–181 (2004). https://doi.org/10.1007/s00245-004-0799-5

    MathSciNet  MATH  Google Scholar 

  10. Chieu, N.H., Lee, G.M.: A relaxed constant positive linear dependence constraint qualification for mathematical programs with equilibrium constraints. J. Optim. Theory Appl. 158(1), 11–32 (2013). https://doi.org/10.1007/s10957-012-0227-y

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  12. Dempe, S.: Foundations of Bilevel Programming. Kluwer, Dordrecht (2002)

    MATH  Google Scholar 

  13. Dempe, S., Dutta, J., Mordukhovich, B. S.: New necessary optimality conditions in optimistic bilevel programming. Optimization 56(5-6), 577–604 (2007). https://doi.org/10.1080/02331930701617551

    MathSciNet  MATH  Google Scholar 

  14. Dempe, S., Franke, S.: The bilevel road pricing problem. Int. J. Comput. Optim. 2(2), 71–92 (2015). https://doi.org/10.12988/ijco.2015.5415

    Google Scholar 

  15. Dempe, S., Franke, S.: On the solution of convex bilevel optimization problems. Comput. Optim. Appl. 63(3), 685–703 (2016). https://doi.org/10.1007/s10589-015-9795-8

    MathSciNet  MATH  Google Scholar 

  16. Dempe, S., Kalashnikov, V., Pérez-Valdéz, G., Kalashnykova, N.: Bilevel Programming Problems - Theory, Algorithms and Applications to Energy Networks. Springer, Berlin (2015)

    MATH  Google Scholar 

  17. Dempe, S., Mordukhovich, B.S., Zemkoho, A.B.: Sensitivity analysis for two-level value functions with applications to bilevel programming. SIAM J. Optim. 22(4), 1309–1343 (2012). https://doi.org/10.1137/110845197

    MathSciNet  MATH  Google Scholar 

  18. Dempe, S., Mordukhovich, B.S., Zemkoho, A.B.: Necessary optimality conditions in pessimistic bilevel programming. Optimization 63(4), 505–533 (2014). https://doi.org/10.1080/02331934.2012.696641

    MathSciNet  MATH  Google Scholar 

  19. Dempe, S., Zemkoho, A.B.: The bilevel programming problem: reformulations, constraint qualifications and optimality conditions. Math. Program. 138 (1), 447–473 (2013). https://doi.org/10.1007/s10107-011-0508-5

    MathSciNet  MATH  Google Scholar 

  20. Fabian, M.J., Henrion, R., Kruger, A.Y., Outrata, J.V.: Error bounds: necessary and sufficient conditions. Set-Valued Variat. Anal. 18 (2), 121–149 (2010). https://doi.org/10.1007/s11228-010-0133-0

    MathSciNet  MATH  Google Scholar 

  21. Fedorov, V.V.: Numerical Maximin Methods. Moscow, Nauka (1979)

    Google Scholar 

  22. Fiacco, A.V.: Optimal Value Continuity and Differential Stability Bounds under the Mangasarian–Fromovitz Constraint Qualification. In: Fiacco, A.V. (ed.) Mathematical Programming with Data Perturbations, vol. 2, pp 65–90. Marcel Dekker, New York (1983)

  23. Fischer, A., Zemkoho, A.B., Zhou, S.: Semismooth Newton-type method for bilevel optimization: global convergence and extensive numerical experiments. arXiv:https://arxiv.org/abs/1912.07079, 1–27 (2019)

  24. Gfrerer, H., Mordukhovich, B.S.: Robinson stability of parametric constraint systems via variational analysis. SIAM J. Optim. 27(1), 438–465 (2017). https://doi.org/10.1137/16M1086881

    MathSciNet  MATH  Google Scholar 

  25. Gfrerer, H., Outrata, J.V.: On Lipschitzian properties of implicit multifunctions. SIAM J. Optim. 26(4), 2160–2189 (2016). https://doi.org/10.1137/15M1052299

    MathSciNet  MATH  Google Scholar 

  26. Guo, L., Lin, G.H.: Notes on some constraint qualifications for mathematical programs with equilibrium constraints. J. Optim. Theory Appl. 156, 600–616 (2013). https://doi.org/10.1007/s10957-012-0084-8

    MathSciNet  MATH  Google Scholar 

  27. Guo, L., Lin, G.H., Ye, J.J.: Second-order optimality conditions for mathematical programs with equilibrium constraints. J. Optim. Theory Appl. 158, 33–64 (2013). https://doi.org/10.1007/s10957-012-0228-x

    MathSciNet  MATH  Google Scholar 

  28. Ioffe, A.D.: Regular points of Lipschitz functions. Trans. Am. Math. Soc. 251, 61–69 (1979). https://doi.org/10.1090/S0002-9947-1979-0531969-6

    MathSciNet  MATH  Google Scholar 

  29. Ioffe, A.D.: Metric regularity and subdifferential calculus. Russ. Math. Surv. 55(3), 501–558 (2000). https://doi.org/10.1070/RM2000v055n03ABEH000292

    MathSciNet  MATH  Google Scholar 

  30. Janin, R.: Directional Derivative of the Marginal Function in Nonlinear Programming. In: Fiacco, A.V. (ed.) Sensitivity, Stability and Parametric Analysis. https://doi.org/10.1007/BFb0121214, vol. 21, pp 110–126. Springer, Berlin (1984)

  31. Klatte, D., Kummer, B.: Stability Properties of Infima and Optimal Solutions of Parametric Optimization Problems. In: Demyanov, V.F., Pallaschke, D. (eds.) Nondifferentiable Optimization: Motivations and Applications, pp 215–229. Springer, Berlin (1985)

  32. Luderer, B., Minchenko, L.I., Satsura, T.: Multivalued Analysis and Nonlinear Programming Problems with Perturbations. Springer Science+Business Media, Dordrecht (2002)

    MATH  Google Scholar 

  33. Mehlitz, P., Minchenko, L.I., Zemkoho, A.B.: A note on partial calmness for bilevel optimization problems with linearly structured lower level. Optim. Lett., 1–15. https://doi.org/10.1007/s11590-020-01636-6 (2020)

  34. Minchenko, L.I., Berezhnov, D.E.: On global partial calmness for bilevel programming problems with linear lower-level problem. In: CEUR Workshop Proceedings. http://ceur-ws.org/Vol-1987/paper60.pdf, vol. 1987 (2017)

  35. Minchenko, L.I., Stakhovski, S.: On relaxed constant rank regularity condition in mathematical programming. Optimization 60(4), 429–440 (2011). https://doi.org/10.1080/02331930902971377

    MathSciNet  MATH  Google Scholar 

  36. Minchenko, L.I., Stakhovski, S.: Parametric nonlinear programming problems under the relaxed constant rank condition. SIAM J. Optim. 21(1), 314–332 (2011). https://doi.org/10.1137/090761318

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  38. Mordukhovich, B.S., Nam, N.M.: Variational stability and marginal functions via generalized differentiation. Math. Oper. Res. 30(4), 800–816 (2005). https://doi.org/10.1287/moor.1050.0147

    MathSciNet  MATH  Google Scholar 

  39. Mordukhovich, B.S., Nam, N.M., Phan, H.M.: Variational analysis of marginal functions with applications to bilevel programming. J. Optim. Theory Appl. 152(3), 557–586 (2012). https://doi.org/10.1007/s10957-011-9940-1

    MathSciNet  MATH  Google Scholar 

  40. Qi, L., Wei, Z.: On the constant positive linear dependence condition and its application to SQP methods. SIAM J. Optim. 10(4), 963–981 (2000). https://doi.org/10.1137/S1052623497326629

    MathSciNet  MATH  Google Scholar 

  41. Robinson, S.M.: Stability theory for systems of inequalities, part II: differentiable nonlinear systems. SIAM J. Numer. Anal. 13(4), 497–513 (1976)

    MathSciNet  MATH  Google Scholar 

  42. Rockafellar, R.T., Wets, R.J.B.: Variational Analysis Grundlehren Der Mathematischen Wissenschaften, vol. 317. Springer, Berlin (1998)

    Google Scholar 

  43. v.Stackelberg, H.: Marktform und Gleichgewicht. Springer, Berlin (1934)

    Google Scholar 

  44. Xu, M., Ye, J.J.: Relaxed constant positive linear dependence constraint qualification and its application to bilevel programs. J. Glob. Optim. 78 (1), 181–205 (2020). https://doi.org/10.1007/s10898-020-00907-x

    MathSciNet  MATH  Google Scholar 

  45. Ye, J.J.: New uniform parametric error bounds. J. Optim. Theory Appl. 98, 197–219 (1998). https://doi.org/10.1023/A:1022649217032

    MathSciNet  MATH  Google Scholar 

  46. Ye, J.J., Zhu, D.L.: Optimality conditions for bilevel programming problems. Optimization 33(1), 9–27 (1995). https://doi.org/10.1080/02331939508844060

    MathSciNet  MATH  Google Scholar 

  47. Ye, J.J., Zhu, D.L.: New necessary optimality conditions for bilevel programs by combining the MPEC and value function approaches. SIAM J. Optim. 20(4), 1885–1905 (2010). https://doi.org/10.1137/080725088

    MathSciNet  MATH  Google Scholar 

  48. Yen, N.D.: Stability of the solution set of perturbed nonsmooth inequality systems and applications. J. Optim. Theory Appl. 93, 199–225 (1997). https://doi.org/10.1023/A:1022662120550

    MathSciNet  MATH  Google Scholar 

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Mehlitz, P., Minchenko, L.I. R-Regularity of Set-Valued Mappings Under the Relaxed Constant Positive Linear Dependence Constraint Qualification with Applications to Parametric and Bilevel Optimization. Set-Valued Var. Anal 30, 179–205 (2022). https://doi.org/10.1007/s11228-021-00578-0

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  • DOI: https://doi.org/10.1007/s11228-021-00578-0

Keywords

  • Bilevel optimization
  • Parametric optimization
  • Partial calmness
  • RCPLD
  • R-regularity

Mathematics Subject Classification 2010

  • 49J53
  • 90C30
  • 90C31