Past exam of the numerical linear algebra and continuous optimisation course of the University of Oxford / 2007 / B / 6 / v

Suppress the iteration subscript. From the quadratic subproblem, $As=-c$ and $g=-Bs+A^{T}y$. ThereforeWhen $|c^{T}y|>0$, the stated bound on $\mu$ makes ${\mu }^{-1}\Vert c{\Vert }_p^p\ge |c^{T}y|$. Consequentlyusing positive definiteness of $B$. Thus every nonzero such SQP step is a descent direction for the penalty function. If $c^{T}y=0$, the quotient bound is unnecessary and the same conclusion follows for every $\mu >0$. A zero step has zero directional derivative and is not a strict descent direction; it corresponds to an already feasible stationary point. This nonzero-step qualification is implicit in the requested conclusion.