  We consider a degenerate nonsmooth and nonconvex optimization problem for
which the standard constraint qualification such as the generalized Mangasarian
Fromovitz constraint qualification (GMFCQ) may not hold. We use smoothing
functions with the gradient consistency property to approximate the nonsmooth
functions and introduce a smoothing sequential quadratic programming (SQP)
algorithm under the exact penalty framework. We show that any accumulation
point of a selected subsequence of the iteration sequence generated by the
smoothing SQP algorithm is a Clarke stationary point, provided that the
sequence of multipliers and the sequence of exact penalty parameters are
bounded. Furthermore, we propose a new condition called the weakly generalized
Mangasarian Fromovitz constraint qualification (WGMFCQ) that is weaker than the
GMFCQ. We show that the extended version of the WGMFCQ guarantees the
boundedness of the sequence of multipliers and the sequence of exact penalty
parameters and thus guarantees the global convergence of the smoothing SQP
algorithm. We demonstrate that the WGMFCQ can be satisfied by bilevel programs
for which the GMFCQ never holds. Preliminary numerical experiments show that
the algorithm is efficient for solving degenerate nonsmooth optimization
problem such as the simple bilevel program.

Optimization and Control