A Note on Optimization Using the Augmented Penalty Function

The augmented penalty function is used to solve optimization problems with constraints and for faster convergence while adopting gradient techniques. In this note, an attempt is made to show that, if x* is an element of S maximizes the function W(x, lambda, K) = f(x) - Sigma(n)(j=1) lambda(j)C(j)(x) - K Sigma(n)(j=1) C(j)(2)(x), then x* maximizes f(x) over all those x is an element of S such that C(j)(x) <= C(j), j = 1, 2, ... , n, under the assumptions that the lambda(j)'s and k are nonnegative, real numbers. Here, W(x, lambda, K), f(x), and C(j)(x), j = 1, 2, ... , n, are real-valued functions and C(j)(x) >= 0 for j = 1, 2, ... , n and for all x. The above result is generalized considering a more general form of the augmented penalty function.