Abstract
We study the convergence of a general perturbation of the Newton method for solving a nonlinear system of equations. As an application, we show that the augmented Lagrangian successive quadratic programming is locally and q-quadratically convergent in the variable x to the solution of an equality constrained optimization problem, under a mild condition on the penalty parameter and the choice of the Lagrange multipliers.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Dennis, J. E., and Schnabel, R. B., Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall, Englewood Cliffs, New Jersey, 1983.
Ortega, J. M., and Rheinboldt, W. C., Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, New York, 1970.
Stewart, G. W., Introduction to Matrix Computations, Academic Press, London, England, 1973.
Bertsekas, D. P., Constrained Optimization and Lagrange Multiplier Methods, Academic Press, New York, New York, 1976.
Fletcher, R., Practical Methods of Optimization, John Wiley and Sons, New York, New York, 1987.
Boggs, P. T., Tolle, J. W., and Wang, P., On the Local Convergence of Quasi-Newton Methods for Constrained Optimization, SIAM Journal on Control and Optimization, Vol. 20, pp. 161–171, 1982.
Haarhoff, P. C., and Buys, J. D., A New Method for the Optimization of a Nonlinear Function Subject to Nonlinear Constraints, Computing Journal, Vol. 13, pp. 178–184, 1970.
Miele, A., Cragg, E. E., and Levy, A. V., On the Method of Multipliers for Mathematical Programming Problems, Part 2, Journal of Optimization Theory and Applications, Vol. 10, pp. 1–33, 1972.
Tapia, R. A., An Introduction to the Algorithm and Theory of Constrained Optimization, Technical Report CRPC-TR90036, Computational and Applied Mathematics Department, Rice University, Houston, Texas, 1970.
Tapia, R. A., Newton's Method for Optimization Problems with Equality Constraints, SIAM Journal on Numerical Analysis, Vol. 11, pp. 874–886, 1974.
Tapia, R. A., Newton's Method for Problems with Equality Constraints, SIAM Journal on Numerical Analysis, Vol. 11, pp. 174–196, 1974.
Tapia, R. A., Diagonalized Multiplier Methods and Quasi-Newton Methods for Constrained Optimization, Journal of Optimization Theory and Applications, Vol. 22, pp. 135–194, 1977.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Cores, D., Tapia, R.A. Perturbation Lemma for the Newton Method with Application to the SQP Newton Method. Journal of Optimization Theory and Applications 97, 271–280 (1998). https://doi.org/10.1023/A:1022622532499
Issue Date:
DOI: https://doi.org/10.1023/A:1022622532499