In this paper, we study the difficult class of optimization problems called the mathematical programs with vanishing constraints or MPVC. Extensive research has been done for MPVC regarding stationary conditions and constraint qualifications using geometric approaches. We use the Fritz John approach for MPVC to derive the M-stationary conditions under weak constraint qualifications. An enhanced Fritz John type stationary condition is also derived for MPVC, which provides the notion of enhanced M-stationarity under a new and weaker constraint qualification: MPVC-generalized quasinormality. We show that this new constraint qualification is even weaker than MPVC-CPLD. A local error bound result is also established under MPVC-generalized quasinormality.

32 pages, to appear in J. Math. Anal. Appl., https://doi.org/10.1016/j.jmaa.2018.11.063