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Zh. Vychisl. Mat. Mat. Fiz., 2005, Volume 45, Number 6, Pages 966–982 (Mi zvmmf636)  

This article is cited in 17 scientific papers (total in 17 papers)

On the analytical and numerical stability of critical Lagrange multipliers

A. F. Izmailov

Faculty of Computational Mathematics and Cybernetics, Moscow State University, Leninskie gory, Moscow, 119992, Russia

Abstract: If the constraint qualification does not hold at a stationary point of a constrained optimization problem, then the corresponding Lagrange multiplier may not be unique. Moreover, in the set of multipliers, one can select special (so-called critical) multipliers possessing certain specific properties that are lacking in the other multipliers. In particular, it is the critical multipliers that are usually stable with respect to small perturbations, and it is the critical multipliers that attract trajectories of Newton's method as applied to the Lagrange system of equations. The present paper is devoted to an analysis of these issues.

Key words: Lagrange multipliers, constrained optimization problems, stability of critical Lagrange multipliers.

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English version:
Computational Mathematics and Mathematical Physics, 2005, 45:6, 930–946

Bibliographic databases:
UDC: 519.626
Received: 16.11.2004

Citation: A. F. Izmailov, “On the analytical and numerical stability of critical Lagrange multipliers”, Zh. Vychisl. Mat. Mat. Fiz., 45:6 (2005), 966–982; Comput. Math. Math. Phys., 45:6 (2005), 930–946

Citation in format AMSBIB
\by A.~F.~Izmailov
\paper On the analytical and numerical stability of critical Lagrange multipliers
\jour Zh. Vychisl. Mat. Mat. Fiz.
\yr 2005
\vol 45
\issue 6
\pages 966--982
\jour Comput. Math. Math. Phys.
\yr 2005
\vol 45
\issue 6
\pages 930--946

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    This publication is cited in the following articles:
    1. M. M. Golishnikov, A. F. Izmailov, “Newton-type methods for constrained optimization with nonregular constraints”, Comput. Math. Math. Phys., 46:8 (2006), 1299–1319  mathnet  crossref  mathscinet
    2. A. F. Izmailov, “Sensitivity of solutions to systems of optimality conditions under the violation of constraint qualifications”, Comput. Math. Math. Phys., 47:4 (2007), 533–554  mathnet  crossref  mathscinet  zmath  elib  elib
    3. Izmailov A.F., Solodov M.V., “Examples of dual behaviour of Newton-type methods on optimization problems with degenerate constraints”, Computational Optimization and Applications, 42:2 (2009), 231–264  crossref  mathscinet  zmath  isi  scopus
    4. Izmailov A.F., Solodov M.V., “On attraction of Newton-type iterates to multipliers violating second-order sufficiency conditions”, Mathematical Programming, 117:1–2 (2009), 271–304  crossref  mathscinet  zmath  isi  elib  scopus
    5. Izmailov A.F., “Solution sensitivity for Karush-Kuhn-Tucker systems with non-unique Lagrange multipliers”, Optimization, 59:5 (2010), 747–775  crossref  mathscinet  zmath  isi  elib  scopus
    6. A. F. Izmailov, E. I. Uskov, “On the application of Newton-type methods to Fritz John optimality conditions”, Comput. Math. Math. Phys., 51:7 (2011), 1114–1127  mathnet  crossref  mathscinet  isi
    7. A. F. Izmailov, E. I. Uskov, “On the influence of the critical Lagrange multipliers on the convergence rate of the multiplier method”, Comput. Math. Math. Phys., 52:11 (2012), 1504–1519  mathnet  crossref  mathscinet  isi  elib  elib
    8. Izmailov A.F., Solodov M.V., “Stabilized Sqp Revisited”, Math. Program., 133:1-2 (2012), 93–120  crossref  mathscinet  zmath  isi  elib  scopus
    9. E. I. Uskov, “On the attraction of Newton’s method to critical Lagrange multipliers”, Comput. Math. Math. Phys., 53:8 (2013), 1099–1112  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    10. Izmailov A.F., Kurennoy A.S., Solodov M.V., “Local Convergence of the Method of Multipliers For Variational and Optimization Problems Under the Noncriticality Assumption”, Comput. Optim. Appl., 60:1 (2015), 111–140  crossref  mathscinet  zmath  isi  scopus
    11. Izmailov A.F., Solodov M.V., “Critical Lagrange Multipliers: What We Currently Know About Them, How They Spoil Our Lives, and What We Can Do About It”, Top, 23:1 (2015), 1–26  crossref  mathscinet  zmath  isi  scopus
    12. Izmailov A.F. Uskov E.I., “Attraction of Newton Method To Critical Lagrange Multipliers: Fully Quadratic Case”, Math. Program., 152:1-2 (2015), 33–73  crossref  mathscinet  zmath  isi  elib  scopus
    13. Izmailov A.F., Uskov E.I., “Subspace-Stabilized Sequential Quadratic Programming”, Comput. Optim. Appl., 67:1 (2017), 129–154  crossref  mathscinet  zmath  isi  scopus
    14. Izmailov A.F. Kurennoy A.S. Solodov M.V., “Critical Solutions of Nonlinear Equations: Local Attraction For Newton-Type Methods”, Math. Program., 167:2 (2018), 355–379  crossref  mathscinet  zmath  isi  scopus
    15. Izmailov A.F. Kurennoy A.S. Solodov M.V., “Critical Solutions of Nonlinear Equations: Stability Issues”, Math. Program., 168:1-2, SI (2018), 475–507  crossref  mathscinet  zmath  isi  scopus
    16. Mordukhovich B.S., Sarabi M.E., “Critical Multipliers in Variational Systems Via Second-Order Generalized Differentiation”, Math. Program., 169:2 (2018), 605–648  crossref  mathscinet  zmath  isi  scopus
    17. Mordukhovich B., Sarabi E., “Criticality of Lagrange Multipliers in Variational Systems”, SIAM J. Optim., 29:2 (2019), 1524–1557  crossref  isi
  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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