Define Variables Define Functions Gradient Search Conjugate Gradient Search One Dimensional Search Demonstration
 Teach Nonlinear Programming Add-in - One Dimensional Direct Search
 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 positive 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 similar manner we find the local maximum of H by moving in the positive direction.
The results of the direct search depend strongly on the starting point. When there are multiple local optima, the methods terminate at an local optimum, but cannot assure the attainment of the global optimum.

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by Paul A. Jensen