×

Second-order multiplier iteration based on a class of nonlinear Lagrangians. (English) Zbl 1470.90134

Summary: Nonlinear Lagrangian algorithm plays an important role in solving constrained optimization problems. It is known that, under appropriate conditions, the sequence generated by the first-order multiplier iteration converges superlinearly. This paper aims at analyzing the second-order multiplier iteration based on a class of nonlinear Lagrangians for solving nonlinear programming problems with inequality constraints. It is suggested that the sequence generated by the second-order multiplier iteration converges superlinearly with order at least two if in addition the Hessians of functions involved in problem are Lipschitz continuous.

MSC:

90C30 Nonlinear programming
90C52 Methods of reduced gradient type
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Hestenes, M. R., Multiplier and gradient methods, Journal of Optimization Theory and Applications, 4, 5, 303-320 (1969) · Zbl 0174.20705 · doi:10.1007/BF00927673
[2] Powell, M. J. D., A method for nonlinear constraints in minimization problems, Optimization, 283-298 (1969), New York, NY, USA: Academic Press, New York, NY, USA · Zbl 0194.47701
[3] Rockafellar, R. T., Augmented Lagrange multiplier functions and duality in nonconvex programming, SIAM Journal on Control, 12, 268-285 (1974) · Zbl 0257.90046 · doi:10.1137/0312021
[4] Bertsekas, D. P., Multiplier methods: a survey, Automatica, 12, 2, 133-145 (1976) · Zbl 0321.49027 · doi:10.1016/0005-1098(76)90077-7
[5] Bertsekas, D. P., On the convergence properties of second-order multiplier methods, Journal of Optimization Theory and Applications, 25, 3, 443-449 (1978) · Zbl 0362.65041 · doi:10.1007/BF00932905
[6] Brusch, R. B., A rapidly convergent methods for equality constrained function minimization, Proceedings of the IEEE Conference on Decision and Control
[7] Fletcher, R., An ideal penalty function for constrained optimization, Nonlinear Programming \(2^″, 121-163 (1975)\), New York, NY, USA: Academic Press, New York, NY, USA · Zbl 0322.90053
[8] Bertsekas, D. P., Constrained Optimization and Lagrange Multiplier Methods (1982), New York, NY, USA: Academic Press, New York, NY, USA · Zbl 0572.90067
[9] Polyak, R. A.; Teboulle, M., Nonlinear rescaling and proximal-like methods in convex optimization, Mathematical Programming, Series B, 76, 2, 265-284 (1997) · Zbl 0882.90106
[10] Polyak, R. A.; Griva, I., Primal-dual nonlinear rescaling method for convex optimization, Journal of Optimization Theory and Applications, 122, 1, 111-156 (2004) · Zbl 1129.90339 · doi:10.1023/B:JOTA.0000041733.24606.99
[11] Griva, I.; Polyak, R. A., Primal-dual nonlinear rescaling method with dynamic scaling parameter update, Mathematical Programming, 106, 2, 237-259 (2006) · Zbl 1134.90494 · doi:10.1007/s10107-005-0603-6
[12] Auslender, A.; Cominetti, R.; Haddou, M., Asymptotic analysis for penalty and barrier methods in convex and linear programming, Mathematics of Operations Research, 22, 1, 43-62 (1997) · Zbl 0872.90067 · doi:10.1287/moor.22.1.43
[13] Ben-Tal, A.; Zibulevsky, M., Penalty/barrier multiplier methods for convex programming problems, SIAM Journal on Optimization, 7, 2, 347-366 (1997) · Zbl 0872.90068 · doi:10.1137/S1052623493259215
[14] Ren, Y.-H.; Zhang, L.-W., The dual algorithm based on a class of nonlinear Lagrangians for nonlinear programming, Proceedings of the 6th World Congress on Intelligent Control and Automation (WCICA ’06) · doi:10.1109/WCICA.2006.1712481
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.