We consider penalty function methods for finding the maximum of a function f over the set \[ S = \{ x \in R^n :g_i (x) \leqq 0{\text{ for }}i = 1, \cdots ,m{\text{ and }}h_j (x) = 0{\text{ for }}j = 1, \cdots ,p\} . \] New conditions, extending earlier work done by Pietrzykowski, are presented under which the penalty function \[ P(x,\mu ) = \mu f(x) - \sum\limits_{i - 1}^m {g_i^ + } (x) - \sum\limits_{j = 1}^p {\left| {h_j (x)} \right|} \] is locally exact. The relationships among the new conditions, Pietrzykowski’s conditions, and Kuhn–tucker constraint qualifications are explored.