Abstract
Recently, Wright proposed a stabilized sequential quadratic programming algorithm for inequality constrained optimization. Assuming the Mangasarian-Fromovitz constraint qualification and the existence of a strictly positive multiplier (but possibly dependent constraint gradients), he proved a local quadratic convergence result. In this paper, we establish quadratic convergence in cases where both strict complementarity and the Mangasarian-Fromovitz constraint qualification do not hold. The constraints on the stabilization parameter are relaxed, and linear convergence is demonstrated when the parameter is kept fixed. We show that the analysis of this method can be carried out using recent results for the stability of variational problems.
Similar content being viewed by others
References
D.P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods, Academic Press: New York, 1982.
F.H. Clarke, “Generalized gradients and applications,” Trans. of the Amer. Math. Soc., vol. 205, pp. 247–262, 1975.
A.L. Dontchev and W.W. Hager, “Lipschitzian stability in nonlinear control and optimization,” SIAM J. Control Optim., vol. 31, pp. 569–603, 1993.
A.L. Dontchev, W.W. Hager, A.B. Poore, and B. Yang, “Optimality, stability and convergence in nonlinear control,” Appl. Math. Optim., vol. 31, pp. 297–326, 1995.
A.L. Dontchev and W.W. Hager, “The Euler approximation in state constrained optimal control,” Department of Mathematics, University of Florida, Gainesville, FL 32611, November, 1997.
A.L. Dontchev and W.W. Hager, “Lipschitzian stability for state constrained nonlinear optimal control,” SIAM J. Control Optim., vol. 35, pp. 698–718, 1998.
A.L. Dontchev and R.T. Rockafellar, “Characterizations of strong regularity for variational inequalities over polyhedral convex sets,” SIAM J. Optim., vol. 6, pp. 1087–1105, 1996.
I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North-Holland: Amsterdam, 1976.
A. Fischer, “Modified Wilson method for nonlinear programs with nonunique multipliers,” Technische Universität Dresden, Germany, February, 1997.
W.W. Hager, “Lipschitz continuity for constrained processes,” SIAM J. Control Optim., vol. 17, pp. 321–338, 1979.
W.W. Hager, “Approximations to the multiplier method,” SIAM J. Numer. Anal., vol. 22, pp. 16–46, 1985.
W.W. Hager, “Convergence of Wright's stabilized SQP algorithm,” Mathematics Department, University of Florida, Gainesville, FL 32611, January, 1997.
W.W. Hager and M.S. Gowda, “Stability in the presence of degeneracy and error estimation,” Mathematics Department, University of Florida, Gainesville, FL 32611, November 22, 1997 (to appear in Math. Programming).
A.J. Hoffman, “On approximate solutions of systems of linear inequalities,” J. Res. Nat. Bur. Standards, vol. 49, pp. 263–265, 1952.
O.L. Mangasarian and S. Fromovitz, “The Fritz-John necessary optimality conditions in the presence of equality and inequality constraints,” J. of Math. Anal. and Appl., vol. 17, pp. 37–47, 1967.
S.M. Robinson, “PerturbedKuhn-Tucker points and rates of convergence for a class of nonlinear-programming algorithms,” Math. Programming, vol. 7, pp. 1–16, 1974.
S.M. Robinson, “Strongly regular generalized equations,” Math. Oper. Res., vol. 5, pp. 43–62, 1980.
R.T. Rockafellar, “The multiplier method of Hestenes and Powell applied to convex programming,” J. Optim. Theory Appl., vol. 12, pp. 555–562, 1973.
S.J. Wright, “Superlinear convergence of a stabilized SQP method to a degenerate solution,” Comput. Optim. Appl., vol. 11, pp. 253–275, 1998.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hager, W.W. Stabilized Sequential Quadratic Programming. Computational Optimization and Applications 12, 253–273 (1999). https://doi.org/10.1023/A:1008640419184
Issue Date:
DOI: https://doi.org/10.1023/A:1008640419184