= Solution
Suppress the iteration subscript. From the quadratic subproblem, $As=-c$ and $g=-Bs+A^Ty$. Therefore
$$
\begin{aligned}
\nabla\Phi(x,\mu)^Ts
&=g^Ts+\mu^{-1}\sum_i c_i^{p-1}(As)_i\\
&=-s^TBs-c^Ty-\mu^{-1}\|c\|_p^p.
\end{aligned}
$$
When $|c^Ty|>0$, the stated bound on $\mu$ makes $\mu^{-1}\|c\|_p^p\geq|c^Ty|$. Consequently
$$
\boxed{\nabla\Phi(x,\mu)^Ts\leq-s^TBs<0\quad\text{if }s\ne0,}
$$
using positive definiteness of $B$. Thus every nonzero such SQP step is a <descent direction> for the penalty function. If $c^Ty=0$, 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.
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