Multi-variable optimization

Abstract

Some of the problems encountered in the minimization of functions of n variables were introduced in Chapter 1. One additional problem, best illustrated by an example, is the enormity of hyperspace. Consider a 10 dimensional unit hypercube in which a search has determined that the volume from the origin to the point ½ in each dimension does not contain the minimum. The volume of the ½ unit cube eliminated from the search is equal to (½)10 ≈ 1/1000. The portion of the unit hypercube which remains is therefore 99.9% of the whole. Any form of complete search is clearly out of the question and attention must be given to the local area of a first estimate to the solution. All methods can therefore guarantee only to find the local minimum.