= Solution
The <softplus function>
$$
h(s)=\log(1+e^s)
$$
is convex because
$$
h''(s)=\frac{e^s}{(1+e^s)^2}\geq0.
$$
Part (b) therefore shows that every $\beta\mapsto h(a_i^T\beta)$ is convex. The <absolute value function> is convex, so $\beta\mapsto|\beta_j|$ is convex for every coordinate $j$. Finally, part (a) says that a finite sum of convex functions is convex. Therefore
$$
Q(\beta)=\sum_{i=1}^n\log(1+e^{a_i^T\beta})+\sum_{j=1}^d|\beta_j|
$$
is convex. The second sum is the <L1 norm> regularizer.
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