A multiplier active-set trust-region algorithm for solving constrained optimization problem. (English) Zbl 1320.90080
Summary: A new trust-region algorithm for solving a constrained optimization problem is introduced. In this algorithm, an active set strategy is used together with multiplier method to convert the computation of the trial step to easy trust-region subproblem similar to this for the unconstrained case. A convergence theory for this algorithm is presented. Under reasonable assumptions, it is shown that the algorithm is globally convergent. In particular, it is shown that, in the limit, a subsequence of the iteration sequence satisfies one of four types of stationary conditions. Namely, the infeasible Mayer-Bliss conditions, Fritz John’s conditions, the infeasible Fritz John’s conditions or KKT conditions.{
}Preliminary numerical experiment on the algorithm is presented. The performance of the algorithm is reported. The numerical results show that our approach is of value and merit further investigation.
Keywords:
constrained optimization; trust region; global convergence; active set; multiplier method; stationary pointsReferences:
| [1] | Conn, A.; Could, N.; Toint, L., (LANCELOT): a fortran package for large-scale nonlinear (release A) optimization, Springer Series in Computational Mathematics, 17 (1992) · Zbl 0761.90087 |
| [2] | Chen, J.; Sun, W.; Sampaio, R., Numerical reserch on the sensitivity of nonmonotone trust region algorithms to their parameters, Computers and Mathematics with Applications, 56, 2932-2940 (2008) · Zbl 1165.65360 |
| [3] | Dennis, J.; El-Alem, M.; Williamson, K., A trust-region approach to nonlinear systems of equalities and inequalities, SIAM Journal on Optimization, 9, 291-315 (1999) · Zbl 0957.65058 |
| [4] | M. El-Alem, A global convergence theory for a class of trust-region algorithms for constrained optimization, Ph.D. thesis, Department of Mathematical Sciences, Rice University, Houston, Texas, 1988.; M. El-Alem, A global convergence theory for a class of trust-region algorithms for constrained optimization, Ph.D. thesis, Department of Mathematical Sciences, Rice University, Houston, Texas, 1988. |
| [5] | El-Alem, M., A global convergence theory for the Dennis, El-Alem, Maciel’s class of trust-region algorithms for constrained optimization without assuming regularity, SIAM Journal on Optimization, 4, 965-989 (1999) · Zbl 0957.65059 |
| [6] | El-Alem, M.; Abd-El-Aziz, M.; El-Bakry, A., A projected Hessian Gauss-Newton algorithm for solving systems of nonlinear equations and inequalities, International Journal of Mathematics and Mathematical Sciences, 25, 6, 397-409 (2001) · Zbl 0982.65066 |
| [7] | B. El-Sobky, Arobust trust-region algorithm for general nonlinear constrained optimization problems, Ph.D. thesis, Department of Mathematics, Alexandria University, Alexandria, Egypt, 1998.; B. El-Sobky, Arobust trust-region algorithm for general nonlinear constrained optimization problems, Ph.D. thesis, Department of Mathematics, Alexandria University, Alexandria, Egypt, 1998. |
| [8] | El-Sobky, B., A global convergence theory for an active trust region algorithm for solving the general nonlinear programming problem, Applied Mathematics and Computation Archive, 144, 1, 127-157 (2003) · Zbl 1110.65049 |
| [9] | Fletcher, R., Practical Methods of Optimization (1987), John Wiley and Sons: John Wiley and Sons Chichester · Zbl 0905.65002 |
| [10] | Fiacco, A.; McCormick, G., Nonlinear Programming: Sequential Unconstrained Minimization Techniques (1968), John Wiley and Sons: John Wiley and Sons New York · Zbl 0193.18805 |
| [11] | Hock, W.; Schittkowski, K., Test examples for nonlinear programming codes, Lecture Notes in Economics and Mathematical Systems, 187 (1981) · Zbl 0452.90038 |
| [12] | Mangasarian, O., Nonlinear Programming (1969), McGraw-Hill Book Company: McGraw-Hill Book Company New York · Zbl 0127.36803 |
| [13] | Moré, J.; Sorensen, D., Computing a trust-region step, SIAM Journal on Scientific and Statistical, Computing, 4, 553-572 (1983) · Zbl 0551.65042 |
| [14] | Shujun, L.; Zhenhai, L., A new trust region filter algorithm, Applied Mathematics and Computation, 204, 485-489 (2008) · Zbl 1167.65034 |
| [15] | Yuan, Y., On the convergence of a new trust region algorithm, Numerical Mathematics, 70, 515-539 (1995) · Zbl 0828.65062 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
