A Genetic Algorithm for Solving Linear-Quadratic Bilevel Program-Ming Problems

2011 ◽  
Vol 186 ◽  
pp. 626-630
Author(s):  
He Cheng Li ◽  
Yu Ping Wang
Keyword(s):  
Genetic Algorithm ◽  
Nonlinear Programming ◽  
Programming Problem ◽  
Original Problem ◽  
Nonlinear Programming Problem ◽  
Linear Quadratic ◽  
Bilevel Programming Problem ◽  
Equivalent Problem ◽  
Bilevel Program ◽  
Two Phases

In this paper, we focus on a special linear-quadratic bilevel programming problem in which the follower’s problem is a convex-quadratic programming, whereas the leader’s functions are linear. At first, based on Karush-Kuhn-Tucher(K-K-T) conditions, the original problem is transformed into an equivalent nonlinear programming problem in which the objective and constraint functions are linear except for the complementary slack conditions. Then, a genetic algorithm is proposed to solve the equivalent problem. In the proposed algorithm, the individuals are encoded in two phases. Finally, the efficiency of the approach is demonstrated by an example.

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.

Minimax Input Shaper Design Using Linear Programming

2008 ◽  
Vol 130 (5) ◽  
Author(s):  
Tarunraj Singh
Keyword(s):  
Linear Programming ◽  
Nonlinear Programming ◽  
Programming Problem ◽  
Linear Programming Problem ◽  
Barycentric Coordinates ◽  
Nonlinear Programming Problem ◽  
Dome A ◽  
Input Shaper ◽  
Maximum Value ◽  
The Cost

The focus of this paper is on the design of robust input shapers where the maximum value of the cost function over the domain of uncertainty is minimized. This nonlinear programming problem is reformulated as a linear programming problem by approximating a n-dimensional hypersphere with multiple hyperplanes (as in a geodesic dome). A recursive technique to approximate a hypersphere to any level of accuracy is developed using barycentric coordinates. The proposed technique is illustrated on the spring-mass-dashpot and the benchmark floating oscillator problem undergoing a rest-to-rest maneuver. It is shown that the results of the linear programming problem are nearly identical to that of the nonlinear programming problem.

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