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.
References
M. R. Hestenes, “Multiplier and gradient methods,” Journal of Optimization Theory and Applications, vol. 4, no. 5, pp. 303–320, 1969. MR0271809 0208.18402 10.1007/BF00927673 M. R. Hestenes, “Multiplier and gradient methods,” Journal of Optimization Theory and Applications, vol. 4, no. 5, pp. 303–320, 1969. MR0271809 0208.18402 10.1007/BF00927673
M. J. D. Powell, “A method for nonlinear constraints in minimization problems,” in Optimization, pp. 283–298, Academic Press, New York, NY, USA, 1969. MR0272403 0194.47701 M. J. D. Powell, “A method for nonlinear constraints in minimization problems,” in Optimization, pp. 283–298, Academic Press, New York, NY, USA, 1969. MR0272403 0194.47701
R. T. Rockafellar, “Augmented Lagrange multiplier functions and duality in nonconvex programming,” SIAM Journal on Control, vol. 12, pp. 268–285, 1974. MR0384163 0257.90046 10.1137/0312021 R. T. Rockafellar, “Augmented Lagrange multiplier functions and duality in nonconvex programming,” SIAM Journal on Control, vol. 12, pp. 268–285, 1974. MR0384163 0257.90046 10.1137/0312021
D. P. Bertsekas, “Multiplier methods: a survey,” Automatica, vol. 12, no. 2, pp. 133–145, 1976. MR0395838 0321.49027 10.1016/0005-1098(76)90077-7 D. P. Bertsekas, “Multiplier methods: a survey,” Automatica, vol. 12, no. 2, pp. 133–145, 1976. MR0395838 0321.49027 10.1016/0005-1098(76)90077-7
D. P. Bertsekas, “On the convergence properties of second-order multiplier methods,” Journal of Optimization Theory and Applications, vol. 25, no. 3, pp. 443–449, 1978. MR508108 0362.65041 10.1007/BF00932905 D. P. Bertsekas, “On the convergence properties of second-order multiplier methods,” Journal of Optimization Theory and Applications, vol. 25, no. 3, pp. 443–449, 1978. MR508108 0362.65041 10.1007/BF00932905
R. B. Brusch, “A rapidly convergent methods for equality constrained function minimization,” in Proceedings of the IEEE Conference on Decision and Control, pp. 80–81, San Diego, Calif, USA, 1973. R. B. Brusch, “A rapidly convergent methods for equality constrained function minimization,” in Proceedings of the IEEE Conference on Decision and Control, pp. 80–81, San Diego, Calif, USA, 1973.
R. Fletcher, “An ideal penalty function for constrained optimization,” in Nonlinear Programming 2$^{{''}}$, pp. 121–163, Academic Press, New York, NY, USA, 1975. MR0389215 R. Fletcher, “An ideal penalty function for constrained optimization,” in Nonlinear Programming 2$^{{''}}$, pp. 121–163, Academic Press, New York, NY, USA, 1975. MR0389215
R. A. Polyak and M. Teboulle, “Nonlinear rescaling and proximal-like methods in convex optimization,” Mathematical Programming, Series B, vol. 76, no. 2, pp. 265–284, 1997. MR1427187 0882.90106 R. A. Polyak and M. Teboulle, “Nonlinear rescaling and proximal-like methods in convex optimization,” Mathematical Programming, Series B, vol. 76, no. 2, pp. 265–284, 1997. MR1427187 0882.90106
R. A. Polyak and I. Griva, “Primal-dual nonlinear rescaling method for convex optimization,” Journal of Optimization Theory and Applications, vol. 122, no. 1, pp. 111–156, 2004. MR2092474 1129.90339 10.1023/B:JOTA.0000041733.24606.99 R. A. Polyak and I. Griva, “Primal-dual nonlinear rescaling method for convex optimization,” Journal of Optimization Theory and Applications, vol. 122, no. 1, pp. 111–156, 2004. MR2092474 1129.90339 10.1023/B:JOTA.0000041733.24606.99
I. Griva and R. A. Polyak, “Primal-dual nonlinear rescaling method with dynamic scaling parameter update,” Mathematical Programming, vol. 106, no. 2, pp. 237–259, 2006. MR2208083 1134.90494 10.1007/s10107-005-0603-6 I. Griva and R. A. Polyak, “Primal-dual nonlinear rescaling method with dynamic scaling parameter update,” Mathematical Programming, vol. 106, no. 2, pp. 237–259, 2006. MR2208083 1134.90494 10.1007/s10107-005-0603-6
A. Auslender, R. Cominetti, and M. Haddou, “Asymptotic analysis for penalty and barrier methods in convex and linear programming,” Mathematics of Operations Research, vol. 22, no. 1, pp. 43–62, 1997. MR1436573 0872.90067 10.1287/moor.22.1.43 A. Auslender, R. Cominetti, and M. Haddou, “Asymptotic analysis for penalty and barrier methods in convex and linear programming,” Mathematics of Operations Research, vol. 22, no. 1, pp. 43–62, 1997. MR1436573 0872.90067 10.1287/moor.22.1.43
A. Ben-Tal and M. Zibulevsky, “Penalty/barrier multiplier methods for convex programming problems,” SIAM Journal on Optimization, vol. 7, no. 2, pp. 347–366, 1997. MR1443623 0872.90068 10.1137/S1052623493259215 A. Ben-Tal and M. Zibulevsky, “Penalty/barrier multiplier methods for convex programming problems,” SIAM Journal on Optimization, vol. 7, no. 2, pp. 347–366, 1997. MR1443623 0872.90068 10.1137/S1052623493259215
Y.-H. Ren and L.-W. Zhang, “The dual algorithm based on a class of nonlinear Lagrangians for nonlinear programming,” in Proceedings of the 6th World Congress on Intelligent Control and Automation (WCICA '06), pp. 934–938, 2006. \endinput Y.-H. Ren and L.-W. Zhang, “The dual algorithm based on a class of nonlinear Lagrangians for nonlinear programming,” in Proceedings of the 6th World Congress on Intelligent Control and Automation (WCICA '06), pp. 934–938, 2006. \endinput
