Non-Linear Unconstrained Optimisation Methods

Abstract

Constrained non-linear optimisation problems are more difficult to solve than unconstrained problems. The previous chapter has described methods that search an unconstrained function to find its lowest value. These search methods are somewhat pedestrian because in general nothing is known about the function being searched. Many trials are needed to build up even a rough picture of how the function behaves. If a set of non-linear constraints or boundaries is added to the function dividing it into feasible and infeasible regions it is no longer sufficient to know merely how the function behaves. The positions, orientations and configurations of the boundaries of the feasible region must also be explored numerically before a constrained minimum can be found. Thus in general, even more trials will be needed to solve a constrained problem than an unconstrained problem. If a non-linear problem has many variables and many constraints the number of trials needed by a numerical direct search process to locate a minimum can be very large and uneconomical. For this reason constrained problems are not usually solved by direct numerical search methods unless they are fairly small problems. Instead various devices, approximations and transformations are used to make the constrained problem easier to solve.