A smoothing-regularization approach to mathematical programs with vanishing constraints

Enhanced Fritz John stationarity, new constraint qualifications and local error bound for mathematical programs with vanishing constraints

“…Here if α = 0, then it contradicts that equation (4), obtained by MPVC-MFCQ, has no nonzero solution with restriction on multipliers given in (3). Hence, α = 0 and then by proper scaling in Fritz John type optimality conditions we have M-stationary conditions at x * .…”

“…(4) also has no nonzero solution with the restriction on the multipliers given in (3). Now, since x * is local minimizer, therefore by the Fritz John type M-stationary condition, there exist a nonzero multiplier (α, λ, µ, η H , η G ) such that…”

On an exact penality result and new constraint qualifications for mathematical programs with vanishing constraints

“…Definition 4.2. Let x * be feasible for (2), then the generalized Mangasarian-Fromovitz constraint qualification (GMFCQ) holds at x * if the following holds…”

An SQP method for mathematical programs with vanishing constraints with strong convergence properties

“…Expectantly, after adding constraint 3 − x 1 − x 2 ≤ 0 to the model (92), to artificially exclude the point (0, 0), unsuitable for the practical application, we reached the point (0, 5), now a global minimizer. For more detailed information about the problem we refer the reader to [7] and [2].…”

“…Recent numerical methods follow different directions. A smoothing-continuation method and a regularization approach for MPCCs are considered in [6,10] and a combination of these techniques, a smoothing-regularization approach for MPVCs is investigated in [2]. In [8,3] the relaxation method has been suggested in order to deal with the inherent difficulties of MPVCs.…”