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

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

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
\Bibitem{Izm05} \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 \mathnet{http://mi.mathnet.ru/zvmmf636} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2193414} \zmath{https://zbmath.org/?q=an:1087.70011} \elib{https://elibrary.ru/item.asp?id=9142380} \transl \jour Comput. Math. Math. Phys. \yr 2005 \vol 45 \issue 6 \pages 930--946 \elib{https://elibrary.ru/item.asp?id=13479077} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
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
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
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
5. Izmailov A.F., “Solution sensitivity for Karush-Kuhn-Tucker systems with non-unique Lagrange multipliers”, Optimization, 59:5 (2010), 747–775
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
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
8. Izmailov A.F., Solodov M.V., “Stabilized Sqp Revisited”, Math. Program., 133:1-2 (2012), 93–120
9. E. I. Uskov, “On the attraction of Newton’s method to critical Lagrange multipliers”, Comput. Math. Math. Phys., 53:8 (2013), 1099–1112
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
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
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
13. Izmailov A.F., Uskov E.I., “Subspace-Stabilized Sequential Quadratic Programming”, Comput. Optim. Appl., 67:1 (2017), 129–154
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
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
16. Mordukhovich B.S., Sarabi M.E., “Critical Multipliers in Variational Systems Via Second-Order Generalized Differentiation”, Math. Program., 169:2 (2018), 605–648
17. Mordukhovich B., Sarabi E., “Criticality of Lagrange Multipliers in Variational Systems”, SIAM J. Optim., 29:2 (2019), 1524–1557
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