Using Newton Method of Optimization - Coursework Example

Using Newton Method of OptimizationConsider the two equations Given that the required volume V is 20m3 and side x is 4m and the cost of the vessel depends on the surface area. The problem is to minimize the surface area to reduce the cost of the vessel.Write surface area S in terms of either h or b as follows: plug the values of volume and side in the volume equation to get an equation in terms of h and b and make h the subject. The resulting equation is . Insert the value of x and substitute for h in the surface area equation to obtain.

Note that it is easier to substitute for h than b in the surface area equation. The surface area equation can also be written as.To optimize S, differentiate it with respect to b to get. Again differentiate it to obtain. The Newton direction is Let, then.  The iterations continue for resulting values of b until d equals zero. When the difference d gets to zero, all iterations ensuing it yield a constant value of b. The constant is the optimal solution. Denoting the subscript of b by k, the table below shows the iterations.

The constant value of b has been obtained after three iterations.Table 1kbk0111.77797121.7888131.78885441.788854