Abstract
For current sequential quadratic programming (SQP) type algorithms, there exist two problems: (i) in order to obtain a search direction, one must solve one or more quadratic programming subproblems per iteration, and the computation amount of this algorithm is very large. So they are not suitable for the large-scale problems; (ii) the SQP algorithms require that the related quadratic programming subproblems be solvable per iteration, but it is difficult to be satisfied. By using ε-active set procedure with a special penalty function as the merit function, a new algorithm of sequential systems of linear equations for general nonlinear optimization problems with arbitrary initial point is presented. This new algorithm only needs to solve three systems of linear equations having the same coefficient matrix per iteration, and has global convergence and local superlinear convergence. To some extent, the new algorithm can overcome the shortcomings of the SQP algorithms mentioned above.
Similar content being viewed by others
References
Han, S. P., Superlinearly convergent variable metric algorithms for general nonlinear programming problems,Math.Prog., 1976, 11:263.
Powell, M. J. D., A fast algorithm for nonlinearly constrained optimization calculations, inNumerical Analysis Proceedings, Dundee 1977 Lecture Notes in Mathematics, Vol 630 (ed. Waston, G. A.). Berlin: Springer-Verlag, 1978, 144.
Chamberlain, R. M., Lemarechal, C., Pedersen, H. C.eta1., The watch-dog technique for forcing convergence in algorithms for constrained optimization,Math. Prog., 1982, 16:1.
Mayne, D. Q., Polak, E., A superlinearly convergent algorithm for constrained optimization problems,Math. Prog. Study, 1982, 16:45.
De, O., Pantoja, J. F. A., Mayne, D. Q., Exact penalty function algorithm with simple updating of the penalty parameter,JOTA, 1991, 69(3):441.
Panier, E. R., Tits, A. L., Herskovits, J. N., A QP-free, globally convergent, locally superlinearly convergent algorithm for inequality constrained optimization,SIAM J. Control and Optimization, 1988, 26(4):788.
Panier, E. R., Tits, A. L., A superlinearly convergent feasible method for the solution of inequality constrained optimization problems,SIAM J. Control and Optimizution, 1987, 25(4):934.
Burke, J. V., Han, S. P., A robust sequential quadratic programming method,Math. Pmg., 1989, 43:277.
Gao, Z. Y., Wu, F.. Lai, Y. L., A superlinearly convergent algorithm of the sequential systems of linear equations for nonlinear optimization problems,Chinese Science Bulletin, 1994, 39(23): 1946.
Gao, Z.Y., He, G.P., Wu, F., A new method for nonlinear optimization problems—Sequential systems of linear equations algorithm, inOperations Rerrarch and Its Applicutions, Lecture Note.s in Operations Research 1 (eds. Du, D.Z., Zhang, X. S., Cheng, K.), World Publishing Corporation, 1995, 64–73.
Powell, M. J. D., Variable metric methods for constrained optimization, inMath. Prog: the State of Art (eds. Bachem, A., Grotschel, M., Korte, B.), Berlin: Springer-Verlag, 1983, 288.
Powell, M. J. D., The convergence of variable metric methods for nonlinear constrained optimization calculations, inNonlin-mr Programming 3 (eds. Mangasarian, O. L., Meyer. R. R., Robinson, S. M.), New York: Academic Press, 1978, 27.
Fukushima, M., A successive quadratic programming algorithm with global and superlinear convergence properties,Math. Prog., 1986, 35:253.
Author information
Authors and Affiliations
Additional information
Project partly supported by the National Natural Science Foundation of China and Tianyuan Foundation of China.
Rights and permissions
About this article
Cite this article
Gao, Z., He, G. & Wu, F. An algorithm of sequential systems of linear equations for nonlinear optimization problems with arbitrary initial point. Sci. China Ser. A-Math. 40, 561–571 (1997). https://doi.org/10.1007/BF02876059
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02876059