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.