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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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Computational Mathematics and Mathematical Physics, 2005, 45:6, 930–946
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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
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This publication is cited in the following articles:
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M. M. Golishnikov, A. F. Izmailov, “Newton-type methods for constrained optimization with nonregular constraints”, Comput. Math. Math. Phys., 46:8 (2006), 1299–1319
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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
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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
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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
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Izmailov A.F., “Solution sensitivity for Karush-Kuhn-Tucker systems with non-unique Lagrange multipliers”, Optimization, 59:5 (2010), 747–775
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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
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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
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Izmailov A.F., Solodov M.V., “Stabilized Sqp Revisited”, Math. Program., 133:1-2 (2012), 93–120
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E. I. Uskov, “On the attraction of Newton’s method to critical Lagrange multipliers”, Comput. Math. Math. Phys., 53:8 (2013), 1099–1112
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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
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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
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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
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Izmailov A.F., Uskov E.I., “Subspace-Stabilized Sequential Quadratic Programming”, Comput. Optim. Appl., 67:1 (2017), 129–154
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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
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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
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Mordukhovich B.S., Sarabi M.E., “Critical Multipliers in Variational Systems Via Second-Order Generalized Differentiation”, Math. Program., 169:2 (2018), 605–648
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Mordukhovich B., Sarabi E., “Criticality of Lagrange Multipliers in Variational Systems”, SIAM J. Optim., 29:2 (2019), 1524–1557
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