Abstract
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.
Highlights
Solving linear programming problems via the simplex method has never had any true competition until Karmarkar presented a new polynomial time method [1,2]
The trust-region strategy can induce strongly global convergence, which is a very important method for solving a nonlinear programming problem and is more robust when it deals with rounding errors
We have included the corresponding results of the penalty–barrier trust-region (PBTR) algorithm against the corresponding numerical results in [38,39]
Summary
Solving linear programming problems via the simplex method has never had any true competition until Karmarkar presented a new polynomial time method [1,2]. Kebbiche et al [11] investigated a projective interior-point algorithm for solving general convex nonlinear problems. Herskovits et al [12] presented an interior-point-based approach for solving bilevel programming problems with convex lower-level problems. Sakineh [15] presented novel interior-point algorithms for solving nonlinear convex optimization problems. The trust-region strategy can induce strongly global convergence, which is a very important method for solving a nonlinear programming problem and is more robust when it deals with rounding errors. It does not require the objective function of the model to be convex.
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