On constraint qualifications in terms of approximate Jacobians for nonsmooth continuous optimization problems

Constraint qualifications in terms of approximate Jacobians are investigated for a nonsmooth constrained optimization problem, in which the involved functions are continuous but not necessarily locally Lipschitz. New constraint qualifications in terms of approximate Jacobians, weaker than the generalized Robinson constraint qualification (GRCQ) in Jeyakumar and Yen [V. Jeyakumar, N.D. Yen, Solution stability of nonsmooth continuous systems with applications to cone-constrained optimization, SIAM J. Optim. 14 5 (2004) 1106–1127], are introduced and some examples are provided to show the utility of constrained qualifications introduced. Since the calmness condition is regarded as the basic condition for optimality conditions, the relationships between the constraint qualifications proposed and the calmness of solution mapping are also studied.