Overcoming Computational Complexity in Nonlinear Optimization

摘要

In this work our main objective is to show how to overcome computational complexity when dealing with nonlinear optimization problems. We consider in particular a nonlinear objective function involving five variables to be maximized subject to four equality constraints. The methodological procedure involves the application of Lagrange multipliers. The first order optimality conditions provide us with the critical values of the problem, while the second order condition is given by the so called Bordered Hessian Matrix. The results of our investigation involved solving highly nonlinear systems of equations involving the Lagrange multipliers, evaluating determinants of very large matrices and finally computing the roots of polynomials of order five. The above difficulties were easily overcome with the help of MathCAD software which proved very efficient in generating numerical solutions to the system of nonlinear equations, evaluating matrix determinants with its symbolic capabilities and computing roots of polynomials with ease. It is shown that for the given problem, only one critical point exists corresponding to a maximum.