  When the objective function is not locally Lipschitz, constraint
qualifications are no longer sufficient for Karush-Kuhn-Tucker (KKT) conditions
to hold at a local minimizer, let alone ensuring an exact penalization. In this
paper, we extend quasi-normality and relaxed constant positive linear
dependence (RCPLD) condition to allow the non-Lipschitzness of the objective
function and show that they are sufficient for KKT conditions to be necessary
for optimality. Moreover, we derive exact penalization results for the
following two special cases. When the non-Lipschitz term in the objective
function is the sum of a composite function of a separable lower
semi-continuous function with a continuous function and an indicator function
of a closed subset, we show that a local minimizer of our problem is also a
local minimizer of an exact penalization problem under a local error bound
condition for a restricted constraint region and a suitable assumption on the
outer separable function. When the non-Lipschitz term is the sum of a
continuous function and an indicator function of a closed subset, we also show
that our problem admits an exact penalization under an extended quasi-normality
involving the coderivative of the continuous function.
