Advertisement

Variational Problems: Local Methods

  • Alexey F. Izmailov
  • Mikhail V. Solodov
Chapter
Part of the Springer Series in Operations Research and Financial Engineering book series (ORFE)

Abstract

In this chapter, we present local analysis of Newton-type algorithms for variational problems, starting with the fundamental Josephy–Newton method for generalized equations. This method is an important extension of classical Newtonian techniques to more general variational problems. For example, as a specific application, the Josephy–Newton method provides a convenient tool for analyzing the sequential quadratic programming algorithm for optimization. We also discuss semismooth Newton methods for complementarity problems, and active-set methods with identifications based on error bounds.

Keywords

Variational Inequality Newton Method Complementarity Problem Sequential Quadratic Programming Linear Complementarity Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 4.
    R. Andreani, E.G. Birgin, J.M. Martínez, M.L. Schuverdt, On Augmented Lagrangian methods with general lower-level constraints. SIAM J. Optim. 18, 1286–1309 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 16.
    A.B. Bakushinskii, A regularization algorithm based on the Newton-Kantorovich method for solving variational inequalities. USSR Comput. Math. Math. Phys. 16, 16–23 (1976)CrossRefGoogle Scholar
  3. 20.
    S.C. Billups, Algorithms for complementarity problems and generalized equations. Ph.D. thesis. Technical Report 95-14. Computer Sciences Department, University of Wisconsin, Madison, 1995Google Scholar
  4. 26.
    J.F. Bonnans, Local analysis of Newton-type methods for variational inequalities and nonlinear programming. Appl. Math. Optim. 29, 161–186 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 44.
    F.H. Clarke, Optimization and Nonsmooth Analysis (Wiley, New York, 1983)zbMATHGoogle Scholar
  6. 48.
    H. Dan, N. Yamashita, M. Fukushima, A superlinearly convergent algorithm for the monotone nonlinear complementarity problem without uniqueness and nondegeneracy conditions. Math. Oper. Res. 27, 743–754 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 49.
    A.N. Daryina, A.F. Izmailov, M.V. Solodov, A class of active-set newton methods for mixed complementarity problems. SIAM J. Optim. 15, 409–429 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 50.
    A.N. Daryina, A.F. Izmailov, M.V. Solodov, Mixed complementarity problems: regularity, error bounds, and Newton-type methods. Comput. Mathem. Mathem. Phys. 44, 45–61 (2004)Google Scholar
  9. 51.
    A.N. Daryina, A.F. Izmailov, M.V. Solodov, Numerical results for a globalized active-set Newton method for mixed complementarity problems. Comput. Appl. Math. 24, 293–316 (2005)zbMATHMathSciNetGoogle Scholar
  10. 52.
    T. De Luca, F. Facchinei, C. Kanzow, A semismooth equation approach to the solution of nonlinear complementarity problems. Math. Program. 75, 407–439 (1996)CrossRefzbMATHGoogle Scholar
  11. 53.
    T. De Luca, F. Facchinei, C. Kanzow, A theoretical and numerical comparison of some semismooth algorithms for complementarity problems. Comput. Optim. Appl. 16, 173–205 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 61.
    A.L. Dontchev, R.T. Rockafellar, Characterizations of strong regularity for variational inequalities over polyhedral convex sets. SIAM J. Optim. 6, 1087–1105 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 62.
    A.L. Dontchev, R.T. Rockafellar, Implicit Functions and Solution Mappings (Springer, New York, 2009)CrossRefzbMATHGoogle Scholar
  14. 68.
    F. Facchinei, J.-S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems (Springer, New York, 2003)Google Scholar
  15. 69.
    F. Facchinei, J. Soares, A new merit function for nonlinear complementarity problems and a related algorithm. SIAM J. Optim. 7, 225–247 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 70.
    F. Facchinei, A. Fischer, C. Kanzow, Regularity properties of a semismooth reformulation of variational inequalities. SIAM J. Optim. 8, 850–869 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 71.
    F. Facchinei, A. Fischer, C. Kanzow, On the accurate identifcation of active constraints. SIAM J. Optim. 9, 14–32 (1999)CrossRefMathSciNetGoogle Scholar
  18. 80.
    M.C. Ferris, C. Kanzow, T.S. Munson, Feasible descent algorithms for mixed complementarity problems. Math. Program. 86, 475–497 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 82.
    A. Fischer, A special Newton-type optimization method. Optimization 24, 296–284 (1992)CrossRefGoogle Scholar
  20. 131.
    A.F. Izmailov, Strongly regular nonsmooth generalized equations. Math. Program. (2013). doi:10.1007/s10107-013-0717-1Google Scholar
  21. 134.
    A.F. Izmailov, A.S. Kurennoy, On regularity conditions for complementarity problems. Comput. Optim. Appl. (2013). doi:10.1007/s10589-013-9604-1Google Scholar
  22. 138.
    A.F. Izmailov, M.V. Solodov, Superlinearly convergent algorithms for solving singular equations and smooth reformulations of complementarity problems. SIAM J. Optim. 13, 386–405 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 139.
    A.F. Izmailov, M.V. Solodov, The theory of 2-regularity for mappings with Lipschitzian derivatives and its applications to optimality conditions. Math. Oper. Res. 27, 614–635 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 140.
    A.F. Izmailov, M.V. Solodov, Karush–Kuhn–Tucker systems: regularity conditions, error bounds and a class of Newton-type methods. Math. Program. 95, 631–650 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 148.
    A.F. Izmailov, M.V. Solodov, Inexact Josephy–Newton framework for generalized equations and its applications to local analysis of Newtonian methods for constrained optimization. Comput. Optim. Appl. 46, 347–368 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 156.
    A.F. Izmailov, A.L. Pogosyan, M.V. Solodov, Semismooth Newton method for the lifted reformulation of mathematical programs with complementarity constraints. Comput. Optim. Appl. 51, 199–221 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 158.
    A.F. Izmailov, A.S. Kurennoy, M.V. Solodov, The Josephy–Newton method for semismooth generalized equations and semismooth SQP for optimization. Set-Valued Variational Anal. 21, 17–45 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 161.
    N.H. Josephy, Newton’s method for generalized equations. Technical Summary Report No. 1965. Mathematics Research Center, University of Wisconsin, Madison, 1979Google Scholar
  29. 162.
    N.H. Josephy, Quasi-Newton methods for generalized equations. Technical Summary Report No. 1966. Mathematics Research Center, University of Wisconsin, Madison, 1979Google Scholar
  30. 163.
    C. Kanzow, Strictly feasible equation-based methods for mixed complementarity problems. Numer. Math. 89, 135–160 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 170.
    D. Klatte, K. Tammer, On the second order sufficient conditions to perturbed C 1, 1 optimization problems. Optimization 19, 169–180 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 179.
    A.S. Lewis, S.J. Wright, Identifying activity. SIAM J. Optim. 21, 597–614 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 209.
    C. Oberlin, S.J. Wright, An accelerated Newton method for equations with semismooth Jacobians and nonlinear complementarity problems. Math. Program. 117, 355–386 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 212.
    J.-S. Pang, S.A. Gabriel, NE/SQP: a robust algorithm for the nonlinear complementarity problem. Math. Program. 60, 295–338 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  35. 221.
    L. Qi, LC1 functions and LC1 optimization problems. Technical Report AMR 91/21. School of Mathematics, The University of New South Wales, Sydney, 1991Google Scholar
  36. 222.
    L. Qi, Convergence analysis of some algorithms for solving nonsmooth equations. Math. Oper. Res. 18, 227–244 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  37. 223.
    L. Qi, Superlinearly convergent approximate Newton methods for LC1 optimization problems. Math. Program. 64, 277–294 (1994)CrossRefzbMATHGoogle Scholar
  38. 224.
    L. Qi, H. Jiang, Semismooth Karush–Kuhn–Tucker equations and convergence analysis of Newton and quasi-Newton methods for solving these equations. Math. Oper. Res. 22, 301–325 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  39. 237.
    R.T. Rockafellar, Computational schemes for solving large-scale problems in extended linear-quadratic programming. Math. Program. 48, 447–474 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  40. 238.
    R.T. Rockafellar, J.B. Wets, Generalized linear-quadratic problems of deterministic and stochastic optimal control in discrete time. SIAM J. Control Optim. 28, 810–922 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  41. 258.
    O. Stein, Lifting mathematical programs with complementarity constraints. Math. Program. 131, 71–94 (2012)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Alexey F. Izmailov
    • 1
  • Mikhail V. Solodov
    • 2
  1. 1.Moscow State UniversityMoscowRussia
  2. 2.Instituto de Matemática Pura e AplicadaRio de JaneiroBrazil

Personalised recommendations