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Linearized M-stationarity Conditions for General Optimization Problems

  • Helmut Gfrerer
Article

Abstract

This paper investigates new first-order optimality conditions for general optimization problems. These optimality conditions are stronger than the commonly used M-stationarity conditions and are in particular useful when the latter cannot be applied because the underlying limiting normal cone cannot be computed effectively. We apply our optimality conditions to a MPEC to demonstrate their practicability.

Keywords

M-stationarity conditions Limiting normal cone Regular normal cone Mathematical programs with equilibrium constraints 

Mathematics Subject Classification (2010)

49J40 49J52 90C 

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Notes

Acknowledgments

The research was partially supported by the Austrian Science Fund (FWF) under grant P29190-N32.

References

  1. 1.
    Adam, L., Henrion, R., Outrata, J.: On M-stationarity conditions in MPECs and the associated qualification conditions. Math. Program. Ser. B 168, 229–259 (2018)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Benko, M., Gfrerer, H.: On estimating the regular normal cone to constraint systems and stationary conditions. Optimization 66, 61–92 (2017)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Benko, M., Gfrerer, H.: New verifiable stationarity concepts for a class of mathematical programs with disjunctive constraints. Optimization 67, 1–23 (2018)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Clarke, F.H.: Necessary Conditions for Nonsmooth Problems in Optimal Control and the Calculus of Variations, Ph.D. Dissertation. University of Washington, Seattle (1973)Google Scholar
  5. 5.
    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)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Dontchev, A.L., Rockafellar, R.T.: Characterizations of strong regularity for variational inequalities over polyhedral convex sets. SIAM J. Optim. 6, 1087–1105 (1996)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Dontchev, A.L., Rockafellar, R.T.: Regularity and conditioning of solution mappings in variational analysis. Set-Valued Var. Anal. 12, 79–109 (2004)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    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)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Gfrerer, H.: First order and second order characterizations of metric subregularity and calmness of constraint set mappings. SIAM J. Optim. 21, 1439–1474 (2011)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    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)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Gfrerer, H., Mordukhovich, B.S.: Complete characterizations of tilt stability in nonlinear programming under weakest qualification conditions. SIAM J. Optim. 25, 2081–2119 (2015)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Gfrerer, H., Outrata, J.V.: On computation of limiting coderivatives of the normal-cone mapping to inequality systems and their applications. Optimization 65, 671–700 (2016)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Gfrerer, H., Outrata, J.V.: On computation of generalized derivatives of the normal-cone mapping and their applications. Math. Oper. Res. 41, 1535–1556 (2016)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Gfrerer, H., Ye, J.J.: New constraint qualifications for mathematical programs with equilibrium constraints via variational analysis. SIAM J. Optim. 27, 842–865 (2017)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Henrion, R., Outrata, J.V.: Calmness of constraint systems with applications. Math. Program., Ser. B 104, 437–464 (2005)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Henrion, R., Outrata, J.V., Surowiec, T.: On the coderivative of normal cone mapping to inequality systems. Nonlinear Anal. 71, 1213–1226 (2009)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Luo, Z.-Q., Pang, J.-S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996)CrossRefMATHGoogle Scholar
  18. 18.
    Mordukhovich, B.S., Outrata, J.V.: On second-order subdifferentials and their applications. SIAM J. Optim. 12, 139–169 (2001)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Outrata, J.V.: Optimality conditions for a class of mathematical programs with equilibrium constraints. Math. Oper. Res. 24, 627–644 (1999)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Scholtes, S.: Convergence properties of a regularization scheme for mathematical programs with complementarity constraints. SIAM J. Optim. 11, 918–936 (2001)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Scheel, H., Scholtes, S.: Mathematical programs with complementarity constraints: stationarity, optimality, and sensitivity. Math. Oper. Res. 25, 1–22 (2000)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Rockafellar, R.T.: Convex analysis. Princeton, New Jersey (1970)CrossRefMATHGoogle Scholar
  23. 23.
    Robinson, S.M.: Generalized equations and their solutions, part I: basic theory. In: Huard, P. (ed.) Point-to-set Maps and Mathematical Programming, vol. 10, pp 128–141. Mathematical Programming Study, North Holland (1979)Google Scholar
  24. 24.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)CrossRefMATHGoogle Scholar
  25. 25.
    Ye, J.J.: Optimality conditions for optimization problems with complementarity constraints. SIAM J. Optim. 9, 374–387 (1999)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Ye, J.J.: Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints. J. Math. Anal. Appl. 307, 350–369 (2005)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Institute of Computational MathematicsJohannes Kepler University (JKU) LinzLinzAustria

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