Second-Order Optimality Conditions for Mathematical Programs with Equilibrium Constraints (original) (raw)

References

  1. Ye, J.J., Zhu, D.L., Zhu, Q.J.: Exact penalization and necessary optimality conditions for generalized bi-level programming problems. SIAM J. Optim. 7, 481–507 (1997)
    Article MathSciNet MATH Google Scholar
  2. Scheel, H.S., Scholtes, S.: Mathematical programs with complementarity constraints: stationarity, optimality, and sensitivity. Math. Oper. Res. 25, 1–22 (2000)
    Article MathSciNet MATH Google Scholar
  3. Ye, J.J.: Constraint qualifications and necessary optimality conditions for optimization problems with variational inequality constraints. SIAM J. Optim. 10, 943–962 (2000)
    Article MathSciNet MATH Google Scholar
  4. Ye, J.J.: Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints. J. Math. Anal. Appl. 307, 350–369 (2005)
    Article MathSciNet MATH Google Scholar
  5. Ye, J.J.: Optimality conditions for optimization problems with complementarity constraints. SIAM J. Optim. 9, 374–387 (1999)
    Article MathSciNet MATH Google Scholar
  6. Ye, J.J., Ye, X.Y.: Necessary optimality conditions for optimization problems with variational inequality constraints. Math. Oper. Res. 22, 977–997 (1997)
    Article MathSciNet MATH Google Scholar
  7. Ye, J.J., Zhang, J.: Enhanced Karush–Kuhn–Tucker condition for mathematical programs with equilibrium constraints. J. Optim. Theory Appl. (to appear)
  8. 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)
    Article MathSciNet MATH Google Scholar
  9. Fukushima, M., Lin, G.H.: Smoothing methods for mathematical programs with equilibrium constraints. In: Proceedings of the ICKS’04, pp. 206–213. IEEE Comput. Soc., Los Alamitos (2004)
    Google Scholar
  10. Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996)
    Book Google Scholar
  11. Outrata, J.V., Kocvara, M., Zowe, J.: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints: Theory, Applications and Numerical Results. Kluwer Academic, Boston (1998)
    Book MATH Google Scholar
  12. Fletcher, R., Leyffer, S., Ralph, D., Scholtes, S.: Local convergence of SQP methods for mathematical programs with equilibrium constraints. SIAM J. Optim. 17, 259–286 (2006)
    Article MathSciNet MATH Google Scholar
  13. Guo, L., Lin, G.H., Ye, J.J.: Stability analysis for parametric mathematical programs with geometric constraints and its applications. SIAM J. Optim. 22, 1151–1176 (2012)
    Article MathSciNet MATH Google Scholar
  14. Hu, X.M., Ralph, D.: Convergence of a penalty method for mathematical programming with equilibrium constraints. J. Optim. Theory Appl. 123, 365–390 (2004)
    Article MathSciNet Google Scholar
  15. Izmailov, A.F., Solodov, M.V.: An active-set Newton method for mathematical programs with complementarity constraints. SIAM J. Optim. 19, 1003–1027 (2008)
    Article MathSciNet MATH Google Scholar
  16. Lin, G.H., Fukushima, M.: A modified relaxation scheme for mathematical programs with complementarity constraints. Ann. Oper. Res. 133, 63–84 (2005)
    Article MathSciNet MATH Google Scholar
  17. Lin, G.H., Guo, L., Ye, J.J.: Solving mathematical programs with equilibrium constraints as constrained equations. Submitted
  18. Scholtes, S.: Convergence properties of a regularization scheme for mathematical programs with complementarity constraints. SIAM J. Optim. 11, 918–936 (2001)
    Article MathSciNet MATH Google Scholar
  19. Izmailov, A.F.: Mathematical programs with complementarity constraints: regularity, optimality conditions and sensitivity. Comput. Math. Math. Phys. 44, 1145–1164 (2004)
    MathSciNet Google Scholar
  20. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley-Interscience, New York (1983)
    MATH Google Scholar
  21. Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)
    MATH Google Scholar
  22. Fiacco, A.V.: Introduction to Sensitivity and Stability Analysis. Academic Press, New York (1983)
    MATH Google Scholar
  23. Fiacco, A.V., McCormick, G.P.: Nonlinear Programming: Sequential Unconstrained Minimization Techniques. Wiley, New York (1968)
    MATH Google Scholar
  24. Ioffe, A.D.: Necessary and sufficient conditions for a local minimum III: second order conditions and augmented duality. SIAM J. Control Optim. 17, 266–288 (1979)
    Article MathSciNet MATH Google Scholar
  25. Andreani, R., Echagüe, C.E., Schuverdt, M.L.: Constant-rank condition and second-order constraint qualification. J. Optim. Theory Appl. 146, 255–266 (2010)
    Article MathSciNet MATH Google Scholar
  26. Gould, N.I.M., Toint, Ph.L.: A note on the convergence of barrier algorithms for second-order necessary points. Math. Program. 85, 433–438 (1999)
    Article MathSciNet MATH Google Scholar
  27. McCormick, G.P.: Second order conditions for constrained minima. SIAM J. Appl. Math. 15, 641–652 (1967)
    Article MathSciNet MATH Google Scholar
  28. Janin, R.: Directional derivative of the marginal function in nonlinear programming. Math. Program. Stud. 21, 110–126 (1984)
    Article MathSciNet MATH Google Scholar
  29. Minchenko, L., Stakhovski, S.: Parametric nonlinear programming problems under the relaxed constant rank condition. SIAM J. Optim. 21, 314–332 (2011)
    Article MathSciNet MATH Google Scholar
  30. Andreani, R., Martinez, J.M., Schuverdt, M.L.: On second-order optimality conditions for nonlinear programming. Optimization 56, 529–542 (2007)
    Article MathSciNet MATH Google Scholar
  31. Qi, L., Wei, Z.X.: On the constant positively linear dependence condition and its application to SQP methods. SIAM J. Optim. 10, 963–981 (2000)
    Article MathSciNet MATH Google Scholar
  32. 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, 473–485 (2005)
    Article MathSciNet MATH Google Scholar
  33. Andreani, R., Haeser, G., Schuverdt, M.L., Silva, J.S.: A relaxed constant positive linear dependence constraint qualification and applications. Math. Program. (2011). doi:10.1007/s10107-011-0456-0
    Google Scholar
  34. Andreani, R., Haeser, G., Schuverdt, M.L., Silva, J.S.: Two new weak constraint qualification and applications. SIAM J. Optim. 22, 1109–1135 (2012)
    Article MathSciNet MATH Google Scholar
  35. Arutyunov, A.V.: Perturbations of extremum problems with constraints and necessary optimality conditions. J. Sov. Math. 54, 1342–1400 (1991)
    Article MATH Google Scholar
  36. Anitescu, M.: Degenerate nonlinear programming with a quadratic growth condition. SIAM J. Optim. 10, 1116–1135 (2000)
    Article MathSciNet MATH Google Scholar
  37. Burke, J.V.: Calmness and exact penalization. SIAM J. Control Optim. 29, 493–497 (1991)
    Article MathSciNet MATH Google Scholar
  38. Guignard, M.: Generalized Kuhn–Tucker conditions for mathematical programs in a Banach space. SIAM J. Control 7, 232–247 (1969)
    Article MathSciNet MATH Google Scholar
  39. Robinson, S.M.: Generalized equations and their solution, part II: applications to nonlinear programming. Math. Program. Stud. 19, 200–221 (1982)
    Article MATH Google Scholar
  40. Robinson, S.M.: Strongly regular generalized equations. Math. Oper. Res. 5, 43–62 (1980)
    Article MathSciNet MATH Google Scholar
  41. Flegel, M.: Constraint qualifications and stationarity concepts for mathematical programs with equilibrium constraints. Ph.D. thesis, University of Würzburg (2005)
  42. Flegel, M.L., Kanzow, C.: On the Guignard constraint qualification for mathematical programs with equilibrium constraints. Optimization 54, 517–534 (2005)
    Article MathSciNet MATH Google Scholar
  43. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I: Basic Theory. Grundlehren der Mathematischen Wissenschaften, vol. 330. Springer, Berlin (2006)
    Google Scholar
  44. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)
    Book MATH Google Scholar
  45. Hoheisel, T., Kanzow, C., Schwartz, A.: Theoretical and numerical comparison of relaxation methods for mathematical programs with complementarity constraints. Math. Program. (2011). doi:10.1007/s10107-011-0488-5
    MATH Google Scholar
  46. Kanzow, C., Schwartz, A.: A new regularization method for mathematical programs with complementarity constraints with strong convergence properties. Submitted
  47. Guo, L., Lin, G.H.: Notes on some constraint qualifications for mathematical programs with equilibrium constraints. J. Optim. Theory Appl. (2012). doi:10.1007/s10957-012-0084-8
    Google Scholar
  48. Bertsekas, D.P., Nedic, A., Ozdaglar, A.E.: Convex Analysis and Optimization. Athena Scientific, Belmont (2003)
    MATH Google Scholar

Download references