Nonlinear Optimization and Excel Solver Nonlinear Optimization Essay

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Nonlinear Optimization and Excel Solver

Nonlinear optimization

Optimization has been considered among the most essential areas of modern applied mathematics, with its applications being in various fields such as in economics, engineering, statistics, finance, medicine and management science. As much as most of the books have described the different aspects, Nonlinear Optimization tend to be the first comprehensive treatment that give opportunity for the graduate students as well as researchers to have an understanding of the modern methods, principles and ideas within a reasonable time, leaving out the use of sacrificing mathematical precision. One of the leading experts, Andrzej Ruszcynski in the optimization of nonlinear stochastic systems goes ahead to integrate the theory and the methods of nonlinear optimization making it to be clear, unified, in addition to mathematically rigorous fashion, and has detailed and easy-to-follow proofs that has been illustrated by use of numerous examples and figures.

Non-Linear Optimization is important for it helps in carrying out minimization or maximization of a function by use of trust region algorithm. In addition, trust region algorithm has been recognized to be very safe and highly efficient, since it converges to a point it provides satisfaction to the first and second order necessary conditions (gradient is zero and Hessian is positive semi-definite) for a local minimum in case the set level of the objective that function below the starting position is bounded, (Fletcher, 1987). Where the point to which algorithm tends to give satisfaction to the second order sufficient condition (Lipschitz in a neighborhood of this point and Hessian is positive definite) the algorithm will converge at second order.

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Usually the algorithm has not been designed for use on functions which have no existence of derivatives or on thousands of variables. Nlm or optim are used for them. Algorithm does best job at local optimization in existence of derivatives. Compared to nlm or optim it is much safer and better behaved, (Nocedal, J. And Wright, S.J., 1999). Its importance is especially where function evaluations tend to be expensive, for it makes the best possible use of each Hessian evaluation, function and gradient.

However, algorithm has not been designed for constrained optimization. It gives opportunity for a restricted domain yet it never converges efficiently to solutions on the boundary of the domain, (Y. Yuan, 1993). This theorems assure rapid convergence to a local optimum in case the level set of the objective function below the starting position has been bounded as well as contained within the interior of the domain of the objective function; in other words, the entire points that are on the boundary of the domain posses higher objective function values as compared to the starting point. Automatically the trust region become adjusted by algorithm in keeping accepted iterates within the interior of the domain, (R. Byrd, R.B. Schnabel and G.A. Shultz, 1987).

OpenSolver

Andrew Mason and students at the Engineering Science department, University of Auckland, NZ, have developed and maintained OpenSolver; it has been described as an Excel VBA add-in that extends built-in Solver of Excel with a highly powerful Linear Programming solver.….....

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