scholarly journals An interior-point trust-region algorithm to solve a nonlinear bilevel programming problem

2022 ◽  
Vol 7 (4) ◽  
pp. 5534-5562
Author(s):  
B. El-Sobky ◽  
◽  
G. Ashry
Keyword(s):  
Programming Problem ◽  
Interior Point ◽  
Bilevel Programming ◽  
Optimal Solution ◽  
Trust Region ◽  
Convergence Theory ◽  
Equilibrium Constraints ◽  
Bilevel Programming Problem ◽  
Trust Region Algorithm ◽  
Starting Point

<abstract><p>In this paper, a nonlinear bilevel programming (NBLP) problem is transformed into an equivalent smooth single objective nonlinear programming (SONP) problem utilized slack variable with a Karush-Kuhn-Tucker (KKT) condition. To solve the equivalent smooth SONP problem effectively, an interior-point Newton's method with Das scaling matrix is used. This method is locally method and to guarantee convergence from any starting point, a trust-region strategy is used. The proposed algorithm is proved to be stable and capable of generating approximal optimal solution to the nonlinear bilevel programming problem.</p> <p>A global convergence theory of the proposed algorithm is introduced and applications to mathematical programs with equilibrium constraints are given to clarify the effectiveness of the proposed approach.</p></abstract>

scholarly journals Trust-Region Based Penalty Barrier Algorithm for Constrained Nonlinear Programming Problems: An Application of Design of Minimum Cost Canal Sections

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1551
Author(s):  
Bothina El-Sobky ◽  
Yousria Abo-Elnaga ◽  
Abd Allah A. Mousa ◽  
Mohamed A. El-Shorbagy
Keyword(s):  
Nonlinear Programming ◽  
Global Convergence ◽  
Programming Problem ◽  
Trust Region ◽  
Convergence Theory ◽  
Benchmark Problems ◽  
Nonlinear Programming Problem ◽  
Barrier Method ◽  
Starting Point ◽  
Constrained Nonlinear Programming

In this paper, a penalty method is used together with a barrier method to transform a constrained nonlinear programming problem into an unconstrained nonlinear programming problem. In the proposed approach, Newton’s method is applied to the barrier Karush–Kuhn–Tucker conditions. To ensure global convergence from any starting point, a trust-region globalization strategy is used. A global convergence theory of the penalty–barrier trust-region (PBTR) algorithm is studied under four standard assumptions. The PBTR has new features; it is simpler, has rapid convergerce, and is easy to implement. Numerical simulation was performed on some benchmark problems. The proposed algorithm was implemented to find the optimal design of a canal section for minimum water loss for a triangle cross-section application. The results are promising when compared with well-known algorithms.

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