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AIMS Mathematics

2022, Volume 7, Issue 9: 16112-16146. doi: 10.3934/math.2022882
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Research article

An active-set with barrier method and trust-region mechanism to solve a nonlinear Bilevel programming problem

  • 1.
    Department of Mathematics and Computer Science, Alexandria University, Faculty of Science, Egypt
  • 2.
    Department of basic science, Tenth of Ramadan City, Higher Technological Institute, Egypt
  • Received: 17 March 2022 Revised: 26 April 2022 Accepted: 14 May 2022 Published: 01 July 2022

  • MSC : 49N10, 49N35, 65K05, 93D22, 93D52

  • Nonlinear Bilevel programming (NBLP) problem is a hard problem and very difficult to be resolved by using the classical method. In this paper, Karush-Kuhn-Tucker (KKT) condition is used with Fischer-Burmeister function to convert NBLP problem to an equivalent smooth single objective nonlinear programming (SONP) problem. An active-set strategy is used with Barrier method and trust-region technique to solve the smooth SONP problem effectively and guarantee a convergence to optimal solution from any starting point. A global convergence theory for the active-set barrier trust-region (ACBTR) algorithm is studied under five standard assumptions. An applications to mathematical programs are introduced to clarify the effectiveness of ACBTR algorithm. The results show that ACBTR algorithm is stable and capable of generating approximal optimal solution to the NBLP problem.

    Citation: B. El-Sobky, G. Ashry, Y. Abo-Elnaga. An active-set with barrier method and trust-region mechanism to solve a nonlinear Bilevel programming problem[J]. AIMS Mathematics, 2022, 7(9): 16112-16146. doi: 10.3934/math.2022882

    Related Papers:

  • Nonlinear Bilevel programming (NBLP) problem is a hard problem and very difficult to be resolved by using the classical method. In this paper, Karush-Kuhn-Tucker (KKT) condition is used with Fischer-Burmeister function to convert NBLP problem to an equivalent smooth single objective nonlinear programming (SONP) problem. An active-set strategy is used with Barrier method and trust-region technique to solve the smooth SONP problem effectively and guarantee a convergence to optimal solution from any starting point. A global convergence theory for the active-set barrier trust-region (ACBTR) algorithm is studied under five standard assumptions. An applications to mathematical programs are introduced to clarify the effectiveness of ACBTR algorithm. The results show that ACBTR algorithm is stable and capable of generating approximal optimal solution to the NBLP problem.


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