A Global Optimization Technique for Solving a Nonlinear Programming Problem with Equality and Inequality Constraints and Its Application to the Bilevel Programming Problem

In this paper, we discuss an exact augumented Lagrangian functions for the non- linear programming problem with both equality and inequality constraints, which is the gen- eration of the augmented Lagrangian function in corresponding reference only for inequality constraints nonlinear programming problem. Under suitable hypotheses, we give the relation- ship between the local and global unconstrained minimizers of the augumented Lagrangian function and the local and global minimizers of the original constrained problem. From the theoretical point of view, the optimality solution of the nonlinear programming with both equality and inequality constraints and the values of the corresponding Lagrangian multipli- ers can be found by the well known method of multipliers which resort to the unconstrained minimization of the augumented Lagrangian function presented in this paper.

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

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.186.626 ◽

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.

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A stochastic approach to global optimization of nonlinear programming problem with many equality constraints

Applied Mathematics and Computation ◽

10.1016/s0096-3003(03)00765-3 ◽

2004 ◽

Vol 155 (1) ◽

pp. 111-120 ◽

Cited By ~ 2

Author(s):

Enmin Feng ◽

Weiyi Qian

Keyword(s):

Global Optimization ◽

Nonlinear Programming ◽

Programming Problem ◽

Stochastic Approach ◽

Equality Constraints ◽

Nonlinear Programming Problem

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

Mathematics ◽

10.3390/math9131551 ◽

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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