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An Effective Method for Solving Nonlinear Programming Problems with Inequality Constraints
Nonlinear programming problems with inequality constraints are a class of optimization problems that arise in various fields like engineering, economics, and science. The objective function and constraints in such problems are nonlinear, which makes them challenging to solve. However, the presence of inequality constraints further complicates these problems. In this paper, we propose an effective method for solving nonlinear programming problems with inequality constraints.
The proposed method is an extension of the classical interior-point method. Interior-point methods are a class of algorithms that work by iteratively approaching the solution of a problem from the interior of the feasible region. The original interior-point method was designed for convex problems with equality constraints. However, several studies have shown that it can also be adapted for nonlinear programming problems with inequality constraints.
Our proposed method uses a primal-dual algorithm that alternates between solving a sequence of linearized problems and updating the directions of step lengths. The linearized problems are derived by approximating the original nonlinear constraints with linear functions. The step-length directions are updated using a modified Newton algorithm.
The proposed method has several advantages over other methods. Firstly, it converges rapidly to high-quality solutions. Secondly, it can handle a wide range of nonlinear programming problems with inequality constraints. Thirdly, it is easy to implement and does not require any special knowledge of nonlinear optimization techniques.
To demonstrate the effectiveness of our method, we present several numerical experiments. We compare our method with other existing methods on a set of benchmark problems. The results show that our method outperforms other methods in terms of solution quality and computational efficiency. Moreover, we test our method on a real-world problem from the chemical engineering domain. The results show that our method can handle real-world problems of practical importance.
In conclusion, we propose an effective method for solving nonlinear programming problems with inequality constraints. The method is an extension of the classical interior-point method and uses a primal-dual algorithm that alternates between solving a sequence of linearized problems and updating the directions of step lengths. The method is easy to implement, converges rapidly to high-quality solutions, and can handle a wide range of nonlinear programming problems with inequality constraints. The proposed method has been demonstrated to outperform existing methods on a set of benchmark problems and a real-world problem from the chemical engineering domain.