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A Globally and Superlinearly Convergent Primal-dual Interior Point Method for General Constrained Optimization

Published online by Cambridge University Press:  05 August 2015

Jianling Li ,
Jian Lv  and
Jinbao Jian
Jianling Li
Affiliation:
College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi, 530004, China
Jian Lv
Affiliation:
School of Mathematics Science, Dalian University of Technology, Dalian, 116024, China
Jinbao Jian*
Affiliation:
School of Mathematics and Information Science, Yulin Normal University; Guangxi Colleges and Universities Key Lab of Complex System Optimization and Big Data Processing, Yulin 537000, China
*
*Email addresses: [email protected] (J.L. Li), [email protected] (J. Lv), [email protected] (J. B. Jian)
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Abstract

In this paper, a primal-dual interior point method is proposed for general constrained optimization, which incorporated a penalty function and a kind of new identification technique of the active set. At each iteration, the proposed algorithm only needs to solve two or three reduced systems of linear equations with the same coefficient matrix. The size of systems of linear equations can be decreased due to the introduction of the working set, which is an estimate of the active set. The penalty parameter is automatically updated and the uniformly positive definiteness condition on the Hessian approximation of the Lagrangian is relaxed. The proposed algorithm possesses global and superlinear convergence under some mild conditions. Finally, some preliminary numerical results are reported.

Keywords

general constrained optimizationprimal-dualactive setglobal convergencesuperlinear convergence.

MSC classification

Secondary: 65K05: Mathematical programming methods 90C30: Nonlinear programming 90C31: Sensitivity, stability, parametric optimization
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 8 , Issue 3 , August 2015 , pp. 313 - 335
Copyright
Copyright © Global-Science Press 2015 

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References

[1]Boggs, P. T. and Tolle, J. W., Sequential quadratic programming, in: Acta No.4, (1995), pp. 1–51. Cambridge University Press, Cambridge, UK.Google Scholar
[2]Bakhtiari, S. and Tits, A. L., A simple primal-dual feasible interior-point method for nonlinear programming with monotone descent, Comput. Optim. Appl., 25 (2003), pp. 17–38.CrossRefGoogle Scholar
[3]Forsgren, A. and Gill, P. E., Primal-dual interior methods for nonconvex nonlinear programming, Siam J. Optim., 8 (1998), pp. 1132–1152.CrossRefGoogle Scholar
[4]Facchinei, F., Fischer, A. and Kanzow, C., On the accurate identification of active constraints, Siam J. Optim., 9 (1998), pp. 14–32.CrossRefGoogle Scholar
[5]Gill, P. E., Murray, W. and Saunders, M. A., SNOPT: An SQP algorithm for large-scale constrained optimization. Siam Review, 47 (2005), pp. 99–131.CrossRefGoogle Scholar
[6]Gao, Z.Y., He, G.P. and Wu, F., Sequential systems of linear equations algorithm for nonlinear opti mization problems with general constraints, J. Optim. Theory Appl., 95 (1997), pp. 371–397.CrossRefGoogle Scholar
[7]Hock, W. and Schittkowski, K., Test examples for nonlinear programming codes. in: Lecture notes in Economics and Mathematical Systems, vol. 187 (1981), Springer-Verlag, Berlin Heidelberg, New York.Google Scholar
[8]Jian, J. B., Quan, R. and Cheng, W. X., A feasible QP-free algorithm combining the interior point method with active set for constrained optimization, Comput. Math. Appl., 58 (2009), pp. 1520–1533.CrossRefGoogle Scholar
[9]Mayne, D. Q. and Polak, E., Feasible direction algorithms for optimization problems with equality and inequality constraints, Math. Program., 11 (1976), pp. 67–80.CrossRefGoogle Scholar
[10]Panier, E. R., Tits, A. L., And Herskovits, J. N., A QP-free, globally convergent, locally superlinearly convergent algorithm for inequality constrained optimization, Siam J. Control Optim., 26 (1988), pp. 788–811.CrossRefGoogle Scholar
[11]Peng, J., Roos, C. and Terlaky, T., New complexity analysis of the primal-dual Newton method for linear optimization, Ann. Oper. Res., 99 (2000), pp. 22–39.CrossRefGoogle Scholar
[12]Qi, H.D., and Qi, L.Q., A new QP-free, globally convergent, locally superlinearly convergent algorithm for inequality constrained optimization, Siam J. Optim., 11 (2000), pp. 113–132.CrossRefGoogle Scholar
[13]Tits, A. L., Wachter, A., Bakhtiari, S., Urban, T.J. and Lawrence, C. T., A primal-dual interior-point method for nonlinear programming with strong global and local convergence properties, Siam J. Optim., 14 (2003), pp. 173–199.CrossRefGoogle Scholar
[14]Wang, Y.L., Chen, L.F. and He, G.P., Sequential systems of linear equations method for general constrained optimization without strict complementarity, J. Comput. Appl. Math., 182 (2005), pp. 447–471.CrossRefGoogle Scholar
[15]Yamashita, H., Yabe, H. and Tanabe, T., A globally and superlinearly convergent primal dual interior point trust region method for large scale constrained optimization, Math. Program., Ser. A, 102 (2005), pp. 111–151.CrossRefGoogle Scholar
[16]Facchinei, F., Lucidi, S., Quadratically and superlinearly convergent for the solution of inequality constrained optimization problem, J. Optim. Theory Appl., 85 (1995), pp. 265–289.CrossRefGoogle Scholar
[17]Yang, Y. F., Li, D. H. and Qi, L. Q.A feasible sequential linear equation method for inequality constrained optimization, Siam J. Optim., 13 (2003), pp. 1222–1244.CrossRefGoogle Scholar