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Operations Research Models and Methods / Methods / Network Methods /
Teach Nonlinear Programming Add-in


One Dimensional Direct Search


When the function to be minimized has a single dimension, such as the one on the left, the direct search routines find the optimum in a single step. The function H(z) is represented on the Excel worksheet as below. The cell labeled other terms has the expression determining the cube of the variable Z times the coefficient in cell J2.

Consideration of this function reveals that for large positive values of z the cubic term dominates and the function value is a large negative number. For large negative values of z, the cubic term again dominates and the function has a large postive value. There is a local minimum and a local maximum nearer to the origin.

We select the Optimize… option from the menu and ask the program to find the minimum of H starting at 0 for the variable z. The program discovers that at z = 0, the gradient is 10 and the normalized gradient is 1. Moving in the direction of the greatest decrease, the line search finds the minimum at z = -9.37783. For a single dimension, the Hessian is simply the second derivative of the function at the stationary point. Since it is positive, the analysis concludes that this is indeed a local minimum.

In a similary manner we find the local maximum of H by moving in the positive direction.

It should be noted that the results of the direct search depend strongly on the starting point.

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Operations Research Models and Methods
by Paul A. Jensen and Jon Bard, University of Texas, Copyright by the Authors