# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2021012

## 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

$^{\star}$ 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.

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
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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
 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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