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doi: 10.3934/jimo.2021012

A new adaptive method to nonlinear semi-infinite programming

Ke Su , , Yumeng Lin , and  Chun Xu

College of Mathematics and Information Science, Hebei University, Hebei Key Laboratory of Machine Learning and Computational Intelligence, Baoding 071002, China

$ ^{\star} $ Corresponding author: Yumeng Lin

Received  March 2020 Revised  October 2020 Published  December 2020

Fund Project: $ ^* $ The first author is supported by the National Natural Science Foundation of China (No. 11101115), the Natural Science Foundation of Hebei Province (grant No. A2018201172), the Key Research Foundation of Education Bureau of Hebei Province (grant No. ZD2015069), the graduate student Innovation ability training project of Hebei University (grant number hbu2020ss043)

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.

Keywords: Semi-infinite programming, trust region, adaptive parameter, convergent properties.
Mathematics Subject Classification: Primary: 90C30; Secondary: 65K05.
Citation: Ke Su, Yumeng Lin, Chun Xu. A new adaptive method to nonlinear semi-infinite programming. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021012
References:
[1]

R. Fletcher and S. Leyffer, Nonlinear programming without a penalty function, Math. Program., 91 (2002), 239-269.  doi: 10.1007/s101070100244.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Table 1.  comparison between Algorithm 1 and algorithm in MATLAB
problem Algorithm 1 Algorithm in MATLAB
Iter CPU $ f(x^{\ast}) $ $ I_{g} $ Iter CPU $ f(x^{\ast}) $ $ I_{g} $
$ 1 $ 14 0.1711 1.8348 25 38 0.1835 2.1126 202
$ 2 $ 25 0.4832 5.3257 31 39 0.7644 5.1293 303
$ 3 $ 33 0.3198 0.1944 43 8 0.5304 0.1945 38
$ 4 $ 26 2.5901 10.7224 27 70 5.9672 11.2252 1010
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problem Algorithm 1 Algorithm in MATLAB
Iter CPU $ f(x^{\ast}) $ $ I_{g} $ Iter CPU $ f(x^{\ast}) $ $ I_{g} $
$ 1 $ 14 0.1711 1.8348 25 38 0.1835 2.1126 202
$ 2 $ 25 0.4832 5.3257 31 39 0.7644 5.1293 303
$ 3 $ 33 0.3198 0.1944 43 8 0.5304 0.1945 38
$ 4 $ 26 2.5901 10.7224 27 70 5.9672 11.2252 1010
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