Article
Full entry |
PDF
(1.3 MB)
Feedback
PDF
(1.3 MB)
Feedback
Keywords:
log-barrier method; Mangasarian–Fromovitz constraint qualification; convergence ofprimal-dual solutions; locally linearized problems; Lipschitz estimates
log-barrier method; Mangasarian–Fromovitz constraint qualification; convergence ofprimal-dual solutions; locally linearized problems; Lipschitz estimates
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 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.
References:
[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 DOI 10.1007/BF01594923 | MR 1098845 | Zbl 0719.90044
[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 DOI 10.1007/BF00249644 | MR 1398265 | Zbl 0862.90120
[4] Fiacco A. V., McCormick G. P.: Nonlinear Programming: Sequential Unconstrained Minimization Techniques. Wiley, New York 1968 MR 0243831 | Zbl 0713.90043
[5] Forsgren A., Gill P. E., Wright M. H.: Interior methods for nonlinear optimization. SIAM Rev. 44 (2002), 525–597 DOI 10.1137/S0036144502414942 | MR 1980444 | Zbl 1028.90060
[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 MR 1752567
[7] Grossmann C., Kaplan A. A.: Strafmethoden und modifizierte Lagrange Funktionen in der nichtlinearen Optimierung. Teubner, Leipzig 1979 MR 0581367 | Zbl 0425.65035
[8] Grossmann C., Schöniger G.: Sensitivität und Anwendbarkeit von Straf–Barriere–Methoden. Z. Angew. Math. Mech. 57 (1977), 419–430 DOI 10.1002/zamm.19770570408 | Zbl 0406.90066
[9] Guddat J., Guerra, F., Nowack D.: On the role of the Mangasarian-Fromovitz constraint qualification for penalty-, exact penalty-, and Lagrange multiplier methods. In: Mathematical programming with data perturbations (A. Fiacco, ed.), (Lecture Notes in Pure and Appl. Mathematics 195). Marcel Dekker, Dordrecht 1997, pp. 159–183 MR 1472270
[10] Güler O.: Limiting behavior of weighted central paths in linear programming. Math. Programming 65A (1994), 347–363 DOI 10.1007/BF01581702 | MR 1296390 | Zbl 0841.90089
[11] Klatte D., Kummer B.: Nonsmooth Equations in Optimization – Regularity, Calculus, Methods and Applications. Kluwer, Dordrecht 2002 MR 1909427 | 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 MR 1336630 | Zbl 0834.90120
[13] Kummer B.: Parametrizations of Kojima’s system and relations to penalty and barrier functions. Math. Programming, Ser. B 76 (1997), 579–592 DOI 10.1007/BF02614399 | MR 1433972 | Zbl 0871.90086
[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 (Translation) MR 0351469 | 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 DOI 10.1287/moor.25.2.179.12227 | MR 1853947 | Zbl 0977.90082
[17] Robinson S. M.: Generalized equations and their solutions. Part II: Applications to nonlinear programming. Math. Programming Stud. 19 (1982), 200–221 DOI 10.1007/BFb0120989 | MR 0669732 | Zbl 0495.90077
[18] Wright S. J.: Superlinear convergence of a stabilized SQP method to a degenerate solution. Comput. Optim. Appl. 11 (1998), 253–275 DOI 10.1023/A:1018665102534 | MR 1651700 | Zbl 0917.90279
[19] Wright S. J.: On the convergence of the Newton/log-barrier method. Math. Programming 90A (2001), 71–100 DOI 10.1007/PL00011421 | MR 1819787 | Zbl 0986.90061
[20] Wright S. J.: Effects of finite-precision arithmetic on interior-point methods for nonlinear programming. SIAM J. Optim. 12 (2001), 36–78 DOI 10.1137/S1052623498347438 | MR 1870586 | Zbl 0994.90139
[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 MR 1926660
Search
Partner of
