Now we turn to the minimization of a function

*of n variables, where and the partial derivatives of**are accessible.*

Steepest Descent or Gradient Method

Steepest Descent or Gradient Method

Let

*be a function of**such that exists for . The gradient of , denoted by , is the vector*(1) .

Illustrations: -

Recall that the gradient vector in (1) points locally in the direction of the greatest rate of increase of . Hence points locally in the direction of greatest decrease . Start at the point and search along the line through in the direction . You will arrive at a point , where a local minimum occurs when the point is constrained to lie on the line . Since partial derivatives are accessible, the minimization process can be executed using either the quadratic or cubic approximation method.

Next we compute and move in the search direction . You will come to , where a local minimum occurs when is constrained to lie on the line . Iteration will produce a sequence, , of points with the property

If then will be a local minimum .

Outline of the Gradient Method

Suppose that has been obtained.

This will produce a value where a local minimum for . The relation

shows that this is a minimum for along the search line .

are the function values and sufficiently close and the distancesmall enough ?

Repeat the process.

**(i)**Evaluate the gradient vector .**(ii)**Compute the search direction .**(iii)**Perform a single parameter minimization of on the interval , where b is large.This will produce a value where a local minimum for . The relation

shows that this is a minimum for along the search line .

**(iv)**Construct the next point .**(v)**Perform the termination test for minimization, i.e.are the function values and sufficiently close and the distancesmall enough ?

Repeat the process.

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