  In this paper we consider a class of structured nonsmooth
difference-of-convex (DC) constrained DC program in which the first convex
component of the objective and constraints is the sum of a smooth and nonsmooth
functions while their second convex component is the supremum of finitely many
convex smooth functions. The existing methods for this problem usually have a
weak convergence guarantee or require a feasible initial point. Inspired by the
recent work (Math Oper. Res. 42(1):95--118, 2017 by Pang et al.), in this paper
we propose two infeasible methods with strong convergence guarantee for the
considered problem. The first one is a penalty method that consists of finding
an approximate D-stationary point of a sequence of penalty subproblems. We show
that any feasible accumulation point of the solution sequence generated by such
a penalty method is a B-stationary point of the problem under a weakest
possible assumption that it satisfies a pointwise Slater constraint
qualification (PSCQ). The second one is an augmented Lagrangian (AL) method
that consists of finding an approximate D-stationary point of a sequence of AL
subproblems. Under the same PSCQ condition as for the penalty method, we show
that any feasible accumulation point of the solution sequence generated by such
an AL method is a B-stationary point of the problem, and moreover, it satisfies
a KKT type of optimality condition for the problem, together with any
accumulation point of the sequence of a set of auxiliary Lagrangian
multipliers. We also propose an efficient successive convex approximation
method for computing an approximate D-stationary point of the penalty and AL
subproblems. Finally, some numerical experiments are conducted to demonstrate
the efficiency of our proposed methods.
