This article is cited in 16 scientific papers (total in 16 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
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.
Lagrange multipliers, constrained optimization problems, stability of critical Lagrange multipliers.
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Computational Mathematics and Mathematical Physics, 2005, 45:6, 930–946
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
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\paper On the analytical and numerical stability of critical Lagrange multipliers
\jour Zh. Vychisl. Mat. Mat. Fiz.
\jour Comput. Math. Math. Phys.
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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
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
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
Izmailov A.F., “Solution sensitivity for Karush-Kuhn-Tucker systems with non-unique Lagrange multipliers”, Optimization, 59:5 (2010), 747–775
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
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
Izmailov A.F., Solodov M.V., “Stabilized Sqp Revisited”, Math. Program., 133:1-2 (2012), 93–120
E. I. Uskov, “On the attraction of Newton’s method to critical Lagrange multipliers”, Comput. Math. Math. Phys., 53:8 (2013), 1099–1112
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
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
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
Izmailov A.F., Uskov E.I., “Subspace-Stabilized Sequential Quadratic Programming”, Comput. Optim. Appl., 67:1 (2017), 129–154
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
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
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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