This thesis considers the problem of minimizing a function f(x) subject to a constraint cp(x) = 0, where f is a scalar quantity, x an n-vector, and cp a q-vector, with q being less than n. The penalty function method is investigated; that is, the vector x is viewed as unconstrained, and the function f(x) is replaced by the penalty function U(x, k) = f(x) + kP(x). Here, the penalty constant k is a positive, scalar quantity, and P(x) = cpT(x)cp(x) is the norm of the constraint error. Crucial to the penalty function method is the prediction of the rate pi = k*/k at which the penalty constant should be increased when shifting from one cycle of the algorithm to the next. Here, k denotes the penalty constant of the present cycle, and k* denotes the penalty constant of the next cycle. In this dissertation, two updating rules are compared: (a) the penalty constant is increased at a constant rate pi = const, and (b) the penalty constant is increased at a variable rate pi = 10j(P/P^), where P is the constraint error at the beginning of a cycle and P is the desired constraint error at convergence. In order to evaluate these updating rules, six numerical examples are investigated. The first example deals with a quadratic function subject to linear constraints; the remaining examples deal with nonquadratic functions subject to nonlinear constraints. Each example is solved with the conjugate gradient algorithm and the modified quasilinearization algorithm for several starting values of the penalty constant. From the numerical experiments, the following conclusions, concerning the number of iterations for convergence, arise: (i) the variable-rate updating rule is superior to the constant-rate updating rule in both the conjugate gradient algorithm and the modified quasilinearization algorithm; and (ii) the above superiority is more pronounced in the conjugate-gradient algorithm than in the modified quasilinearization algorithm.