difficulty associated with steepest descent; any optimization technique can be plagued by

this problem [4].

 Monte Carlo Integer Programming

 Exhaustive search methods for solving optimization problems will produce the

optimal solution, however, as the search space increases it is doubtful that the solution

will be determined in a reasonable amount of time. A more efficient method might

require checking only a sample of possible solutions and then selecting the best solution

out of the sample population. Therefore, random samples of feasible solutions are chosen

and the corresponding (maximum or minimum) best solution from the random sample is

determined. This method is known as Monte Carlo integer programming. There is some

concern regarding the 'goodness' of the answer obtained using this method mainly

attributed to the random nature in which the samples are selected [3]. However, based on

the statistics of the objective function, the random samples follow the probability

distribution of the possible solutions for the integer-programming problem.

Suppose the following function is to be maximized:

 P = x,2 + X2 +10X32 +5x42 +6x52 -3x1 2 +3X3 -2X4 +X5 (3.9)

where 0 < x, < 99 (i=1...5)

There are five variables in the above equation with values ranging from 0 to 99. This

means there are a total of 1005 possible points to check (10,000,000,000). Although

computers are becoming faster, some computers may be burdened by this many

calculations. The Monte Carlo technique will check a random sample of the 1005 points

for a feasible solution. For example, a random sample of one million points is evaluated

and the point with the maximum value will be deemed the solution to the problem.