-
Previous Article
Solving the facility location and fixed charge solid transportation problem
- JIMO Home
- This Issue
-
Next Article
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 |
[1] |
Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1779-1799. doi: 10.3934/dcdss.2020454 |
[2] |
Shu-Yu Hsu. Existence and properties of ancient solutions of the Yamabe flow. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 91-129. doi: 10.3934/dcds.2018005 |
[3] |
M. R. S. Kulenović, J. Marcotte, O. Merino. Properties of basins of attraction for planar discrete cooperative maps. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2721-2737. doi: 10.3934/dcdsb.2020202 |
[4] |
Ardeshir Ahmadi, Hamed Davari-Ardakani. A multistage stochastic programming framework for cardinality constrained portfolio optimization. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 359-377. doi: 10.3934/naco.2017023 |
[5] |
Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399 |
[6] |
Charlene Kalle, Niels Langeveld, Marta Maggioni, Sara Munday. Matching for a family of infinite measure continued fraction transformations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (11) : 6309-6330. doi: 10.3934/dcds.2020281 |
[7] |
Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327 |
[8] |
Seung-Yeal Ha, Myeongju Kang, Bora Moon. Collective behaviors of a Winfree ensemble on an infinite cylinder. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2749-2779. doi: 10.3934/dcdsb.2020204 |
[9] |
Wei Liu, Pavel Krejčí, Guoju Ye. Continuity properties of Prandtl-Ishlinskii operators in the space of regulated functions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3783-3795. doi: 10.3934/dcdsb.2017190 |
[10] |
Marcelo Messias. Periodic perturbation of quadratic systems with two infinite heteroclinic cycles. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1881-1899. doi: 10.3934/dcds.2012.32.1881 |
[11] |
Zhihua Zhang, Naoki Saito. PHLST with adaptive tiling and its application to antarctic remote sensing image approximation. Inverse Problems & Imaging, 2014, 8 (1) : 321-337. doi: 10.3934/ipi.2014.8.321 |
[12] |
Murat Uzunca, Ayşe Sarıaydın-Filibelioǧlu. Adaptive discontinuous galerkin finite elements for advective Allen-Cahn equation. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 269-281. doi: 10.3934/naco.2020025 |
[13] |
Rabiaa Ouahabi, Nasr-Eddine Hamri. Design of new scheme adaptive generalized hybrid projective synchronization for two different chaotic systems with uncertain parameters. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2361-2370. doi: 10.3934/dcdsb.2020182 |
[14] |
V. Vijayakumar, R. Udhayakumar, K. Kavitha. On the approximate controllability of neutral integro-differential inclusions of Sobolev-type with infinite delay. Evolution Equations & Control Theory, 2021, 10 (2) : 271-296. doi: 10.3934/eect.2020066 |
2019 Impact Factor: 1.366
Tools
Metrics
Other articles
by authors
[Back to Top]