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Convergence of primal-dual solutions for the nonconvex log-barrier method without LICQ. (English) Zbl 1249.90252
Summary: This paper characterizes completely the behavior of the logarithmic barrier method under a standard second-order condition, strict (multivalued) complementarity and MFCQ at a local minimizer. We present direct proofs, based on certain key estimates and a few well-known facts on linear and parametric programming, in order to verify existence and Lipschitzian convergence of local primal-dual solutions without applying additionally technical tools arising from Newton-techniques.

MSC:
90C30 Nonlinear programming
65K10 Numerical optimization and variational techniques
49K40 Sensitivity, stability, well-posedness
49M37 Numerical methods based on nonlinear programming
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[1] Adler I., Monteiro R. D. C.: Limiting behavior of the affine scaling continuous trajectories for linear programming problems. Math. Programming 50 (1991), 29-51 · Zbl 0719.90044 · doi:10.1007/BF01594923
[2] Bank B., Guddat J., Klatte D., Kummer, B., Tammer K.: Non-linear Parametric Optimization. Akademie-Verlag, Berlin 1982 · Zbl 0502.49002
[3] El-Bakry A. S., Tapia R. A., Zhang Y.: On the convergence rate of Newton interior-point methods in the absence of strict complementarity. Comput. Optim. Appl. 6 (1996), 157-167 · Zbl 0862.90120 · doi:10.1007/BF00249644
[4] Fiacco A. V., McCormick G. P.: Nonlinear Programming: Sequential Unconstrained Minimization Techniques. Wiley, New York 1968 · Zbl 0713.90043
[5] Forsgren A., Gill P. E., Wright M. H.: Interior methods for nonlinear optimization. SIAM Rev. 44 (2002), 525-597 · Zbl 1028.90060 · doi:10.1137/S0036144502414942
[6] Grossmann C.: Convergence analysis for penalty/barrier path-following in linearly constrained convex programming. Discuss. Math.: Differential Inclusions, Control and Optimization 20 (2000), 7-26 · Zbl 0980.90085 · doi:10.7151/dmdico.1001
[7] Grossmann C., Kaplan A. A.: Strafmethoden und modifizierte Lagrange Funktionen in der nichtlinearen Optimierung. Teubner, Leipzig 1979 · Zbl 0425.65035
[8] Grossmann C., Schöniger G.: Sensitivität und Anwendbarkeit von Straf-Barriere-Methoden. Z. Angew. Math. Mech. 57 (1977), 419-430 · Zbl 0406.90066 · doi:10.1002/zamm.19770570408
[9] Guddat J., Guerra, F., Nowack D.: On the role of the Mangasarian-Fromovitz constraint qualification for penalty-, exact penalty-, and Lagrange multiplier methods. Mathematical programming with data perturbations (A. Fiacco, (Lecture Notes in Pure and Appl. Mathematics 195). Marcel Dekker, Dordrecht 1997, pp. 159-183 · Zbl 0888.90136
[10] Güler O.: Limiting behavior of weighted central paths in linear programming. Math. Programming 65A (1994), 347-363 · Zbl 0841.90089 · doi:10.1007/BF01581702
[11] Klatte D., Kummer B.: Nonsmooth Equations in Optimization - Regularity, Calculus, Methods and Applications. Kluwer, Dordrecht 2002 · Zbl 1173.49300
[12] Kummer B.: On solvability and regularity of a parametrized version of optimality conditions. Z. Oper. Res.: Math. Meth. Oper. Res. 45 (1995), 215-230 · Zbl 0834.90120 · doi:10.1007/BF01432656
[13] Kummer B.: Parametrizations of Kojima’s system and relations to penalty and barrier functions. Math. Programming, Ser. B 76 (1997), 579-592 · Zbl 0871.90086 · doi:10.1007/BF02614399
[14] Nožička F., Guddat J., Hollatz, H., Bank B.: Theorie der linearen parametrischen Optimierung. Akademie-Verlag, Berlin 1974 · Zbl 0284.90053
[15] Owen G.: Spieltheorie. Springer-Verlag, Berlin 1971 · Zbl 0222.90048
[16] Ralph D., Wright S. J.: Superlinear convergence of an interior point method despite dependent constraints. Math. Oper. Res. 25 (2000), 179-194 · Zbl 0977.90082 · doi:10.1287/moor.25.2.179.12227
[17] Robinson S. M.: Generalized equations and their solutions. Part II: Applications to nonlinear programming. Math. Programming Stud. 19 (1982), 200-221 · Zbl 0495.90077 · doi:10.1007/BFb0120989
[18] Wright S. J.: Superlinear convergence of a stabilized SQP method to a degenerate solution. Comput. Optim. Appl. 11 (1998), 253-275 · Zbl 0917.90279 · doi:10.1023/A:1018665102534
[19] Wright S. J.: On the convergence of the Newton/log-barrier method. Math. Programming 90A (2001), 71-100 · Zbl 0986.90061 · doi:10.1007/s101070100195
[20] Wright S. J.: Effects of finite-precision arithmetic on interior-point methods for nonlinear programming. SIAM J. Optim. 12 (2001), 36-78 · Zbl 0994.90139 · doi:10.1137/S1052623498347438
[21] Wright S. J., Orban D.: Properties of the Log-barrier Function on Degenerate Nonlinear Programs. Preprint ANL/MCS-P772-0799, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne 1999, revised May 2001
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