  We propose an SQP algorithm for mathematical programs with vanishing
constraints which solves at each iteration a quadratic program with linear
vanishing constraints. The algorithm is based on the newly developed concept of
$\mathcal Q$-stationarity [5]. We demonstrate how $\mathcal Q_M$-stationary
solutions of the quadratic program can be obtained. We show that all limit
points of the sequence of iterates generated by the basic SQP method are at
least M-stationary and by some extension of the method we also guarantee the
stronger property of $\mathcal Q_M$-stationarity of the limit points.
