The Penalty Function Method

Very often we are interested in minimizing a (“target”) function1; i.e., in finding values of the variables that ensure a minimum of the function when some constraints are satisfied. Just imagine a strange Smoky Mountains hiking trip: we want to find the point of the lowest ground elevation provided that we hike along a straight line from town A to B. Suppose that the target function for minimization (that corresponds to the elevation of the ground in the Smoky Mountains region) is the function f (x1, x2, . . . , xn+m), but the variables xi have to fulfill m equations (“constraints”):