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On the $ BMAP_1, BMAP_2/PH/g, c $ retrial queueing system
A new adaptive method to nonlinear semi-infinite programming
College of Mathematics and Information Science, Hebei University, Hebei Key Laboratory of Machine Learning and Computational Intelligence, Baoding 071002, China |
In this paper, we propose a new adaptive method for solving nonlinear semi-infinite programming(SIP). In the presented method, the continuous infinite inequality constraints are transformed into equivalent equality constraints in integral form. Based on penalty method and trust region strategy, we propose a modified quadratic subproblem, in which an adaptive parameter is considered. The acceptable criterion of the trial point is adjustable according to the value of this adaptive parameter and the improvements that made by the current iteration. Compared with the existing methods, our method is more flexible. Under some reasonable conditions, the convergent properties of the proposed algorithm are proved. The numerical results are reported in the end.
References:
[1] |
R. Fletcher and S. Leyffer,
Nonlinear programming without a penalty function, Math. Program., 91 (2002), 239-269.
doi: 10.1007/s101070100244. |
[2] |
R. Fletcher, S. Leyffer and P. L. Toint,
On the global convergence of a filter-SQP algorithm, SIAM J. Optim., 13 (2002), 44-59.
doi: 10.1137/S105262340038081X. |
[3] |
G. Gramlich, R. Hettich and E. W. Sachs,
Local convergence of SQP methods in semi-infinite programming, SIAM J. Optim., 5 (1995), 641-658.
doi: 10.1137/0805031. |
[4] |
L. S. Jennings and K. L. Teo, A computational algorithm for functional inequality constrained optimization problems, Automatica J. IFAC, 26 (1990), 371-375.
doi: 10.1016/0005-1098(90)90131-Z. |
[5] |
J.-B. Jian, Q.-J. Xu and D.-L. Han,
A norm-relaxed method of feasible directions for finely discretized problems from semi-infinite programming, European J. Oper. Res., 186 (2008), 41-62.
doi: 10.1016/j.ejor.2007.01.026. |
[6] |
D. Li and D. Zhu,
An affine scaling interior trust-region method combining with line search filter technique for optimization subject to bounds on variables, Numer. Algorithms, 77 (2018), 1159-1182.
doi: 10.1007/s11075-017-0357-2. |
[7] |
Y. Liu, K. L. Teo and S. Y. Wu,
A new quadratic semi-infinite programming algorithm based on dual parametrization, J. Global Optim., 29 (2004), 401-413.
doi: 10.1023/B:JOGO.0000047910.80739.95. |
[8] |
J. Lv, L.-P. Pang and F.-Y. Meng,
A proximal bundle method for constrained nonsmooth nonconvex optimization with inexact information, J. Global Optim., 70 (2018), 517-549.
doi: 10.1007/s10898-017-0565-2. |
[9] |
Y.-G. Ou,
A filter trust region method for solving semi-infinite programming problems, J. Appl. Math. Comput., 29 (2009), 311-324.
doi: 10.1007/s12190-008-0132-6. |
[10] |
L. Pang and D. Zhu,
A line search filter-SQP method with Lagrangian function for nonlinear inequality constrained optimization, Jpn. J. Ind. Appl. Math., 34 (2017), 141-176.
doi: 10.1007/s13160-017-0236-1. |
[11] |
R. Reemtsen,
Discretization methods for the solution of semi-infinite programming problems, J. Optim. Theory Appl., 71 (1991), 85-103.
doi: 10.1007/BF00940041. |
[12] |
Y. Tanaka, M. Fukushima and T. Ibaraki,
A globally convergent SQP method for semi-infinite nonlinear optimization, J. Comput. Appl. Math., 23 (1988), 141-153.
doi: 10.1016/0377-0427(88)90276-2. |
[13] |
K. L. Teo, V. Rehbock and L. S. Jennings,
A new computational algorithm for functional inequality constrained optimization problems, Automatica J. IFAC, 29 (1993), 789-792.
doi: 10.1016/0005-1098(93)90076-6. |
[14] |
K. L. Teo, X. Q. Yang and L. S. Jennings,
Computational discretization algorithms for functional inequality constrained optimization, Ann. Oper. Res., 98 (2000), 215-234.
doi: 10.1023/A:1019260508329. |
[15] |
S.-Y. Wu, D. H. Li, and L. Qi and G. Zhou,
An iterative method for solving KKT system of the semi-infinite programming, Optim. Methods Softw., 20 (2005), 629-643.
doi: 10.1080/10556780500094739. |
[16] |
C. Yu, K. L. Teo, L. Zhang and Y. Bai,
A new exact penalty function method for continuous inequality constrained optimization problems, J. Ind. Manag. Optim., 6 (2010), 895-910.
doi: 10.3934/jimo.2010.6.895. |
[17] |
L. Zhang, S.-Y. Wu and M. A. López,
A new exchange method for convex semi-infinite programming, SIAM J. Optim., 20 (2010), 2959-2977.
doi: 10.1137/090767133. |
show all references
References:
[1] |
R. Fletcher and S. Leyffer,
Nonlinear programming without a penalty function, Math. Program., 91 (2002), 239-269.
doi: 10.1007/s101070100244. |
[2] |
R. Fletcher, S. Leyffer and P. L. Toint,
On the global convergence of a filter-SQP algorithm, SIAM J. Optim., 13 (2002), 44-59.
doi: 10.1137/S105262340038081X. |
[3] |
G. Gramlich, R. Hettich and E. W. Sachs,
Local convergence of SQP methods in semi-infinite programming, SIAM J. Optim., 5 (1995), 641-658.
doi: 10.1137/0805031. |
[4] |
L. S. Jennings and K. L. Teo, A computational algorithm for functional inequality constrained optimization problems, Automatica J. IFAC, 26 (1990), 371-375.
doi: 10.1016/0005-1098(90)90131-Z. |
[5] |
J.-B. Jian, Q.-J. Xu and D.-L. Han,
A norm-relaxed method of feasible directions for finely discretized problems from semi-infinite programming, European J. Oper. Res., 186 (2008), 41-62.
doi: 10.1016/j.ejor.2007.01.026. |
[6] |
D. Li and D. Zhu,
An affine scaling interior trust-region method combining with line search filter technique for optimization subject to bounds on variables, Numer. Algorithms, 77 (2018), 1159-1182.
doi: 10.1007/s11075-017-0357-2. |
[7] |
Y. Liu, K. L. Teo and S. Y. Wu,
A new quadratic semi-infinite programming algorithm based on dual parametrization, J. Global Optim., 29 (2004), 401-413.
doi: 10.1023/B:JOGO.0000047910.80739.95. |
[8] |
J. Lv, L.-P. Pang and F.-Y. Meng,
A proximal bundle method for constrained nonsmooth nonconvex optimization with inexact information, J. Global Optim., 70 (2018), 517-549.
doi: 10.1007/s10898-017-0565-2. |
[9] |
Y.-G. Ou,
A filter trust region method for solving semi-infinite programming problems, J. Appl. Math. Comput., 29 (2009), 311-324.
doi: 10.1007/s12190-008-0132-6. |
[10] |
L. Pang and D. Zhu,
A line search filter-SQP method with Lagrangian function for nonlinear inequality constrained optimization, Jpn. J. Ind. Appl. Math., 34 (2017), 141-176.
doi: 10.1007/s13160-017-0236-1. |
[11] |
R. Reemtsen,
Discretization methods for the solution of semi-infinite programming problems, J. Optim. Theory Appl., 71 (1991), 85-103.
doi: 10.1007/BF00940041. |
[12] |
Y. Tanaka, M. Fukushima and T. Ibaraki,
A globally convergent SQP method for semi-infinite nonlinear optimization, J. Comput. Appl. Math., 23 (1988), 141-153.
doi: 10.1016/0377-0427(88)90276-2. |
[13] |
K. L. Teo, V. Rehbock and L. S. Jennings,
A new computational algorithm for functional inequality constrained optimization problems, Automatica J. IFAC, 29 (1993), 789-792.
doi: 10.1016/0005-1098(93)90076-6. |
[14] |
K. L. Teo, X. Q. Yang and L. S. Jennings,
Computational discretization algorithms for functional inequality constrained optimization, Ann. Oper. Res., 98 (2000), 215-234.
doi: 10.1023/A:1019260508329. |
[15] |
S.-Y. Wu, D. H. Li, and L. Qi and G. Zhou,
An iterative method for solving KKT system of the semi-infinite programming, Optim. Methods Softw., 20 (2005), 629-643.
doi: 10.1080/10556780500094739. |
[16] |
C. Yu, K. L. Teo, L. Zhang and Y. Bai,
A new exact penalty function method for continuous inequality constrained optimization problems, J. Ind. Manag. Optim., 6 (2010), 895-910.
doi: 10.3934/jimo.2010.6.895. |
[17] |
L. Zhang, S.-Y. Wu and M. A. López,
A new exchange method for convex semi-infinite programming, SIAM J. Optim., 20 (2010), 2959-2977.
doi: 10.1137/090767133. |
problem | Algorithm 1 | Algorithm in MATLAB | |||||||
Iter | CPU | Iter | CPU | ||||||
|
14 | 0.1711 | 1.8348 | 25 | 38 | 0.1835 | 2.1126 | 202 | |
25 | 0.4832 | 5.3257 | 31 | 39 | 0.7644 | 5.1293 | 303 | ||
33 | 0.3198 | 0.1944 | 43 | 8 | 0.5304 | 0.1945 | 38 | ||
26 | 2.5901 | 10.7224 | 27 | 70 | 5.9672 | 11.2252 | 1010 |
problem | Algorithm 1 | Algorithm in MATLAB | |||||||
Iter | CPU | Iter | CPU | ||||||
|
14 | 0.1711 | 1.8348 | 25 | 38 | 0.1835 | 2.1126 | 202 | |
25 | 0.4832 | 5.3257 | 31 | 39 | 0.7644 | 5.1293 | 303 | ||
33 | 0.3198 | 0.1944 | 43 | 8 | 0.5304 | 0.1945 | 38 | ||
26 | 2.5901 | 10.7224 | 27 | 70 | 5.9672 | 11.2252 | 1010 |
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