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, ifx* ∈S maximizes the function $$W(x,\lambda ,{\rm K}) = f(x) - \sum\limits_{j = 1}^n {\lambda _j C_j (x)} - K\sum\limits_{j = 1}^n {C_j ^2 (x)} ,$$ thenx* maximizesf(x) over all thosex ∈S such that $$C_j (x) \leqslant C_j ,j = 1,2, \ldots ,n,$$ under the assumptions that the λ _j 's andk are nonnegative, real numbers. Here,W(x, λ,K),f(x), andC _j (x),j=1, 2,...,n, are real-valued functions andC _j (x) ≥ 0 forj=1, 2,...,n and for allx. The above result is generalized considering a more general form of the augmented penalty function.