Minimize $ {L}_{p} $ Norm Regularized with a Linear Term (Conjugate Function of the Norm Function) The problem is given as following:
$$ \min_{x} {a}^{T} x + \lambda \left\| x \right\|_{p} $$
Namely minimizing a ${L}_{p} $ norm term regularized by a linear term.
The above form occurs repeatedly on the dual forms of convex optimization problems as it is related to the Conjugate Function.
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I'll take $\lambda = 1$ for simplicity. Let $f(x) = \| x \|_p$. Note that
\begin{equation}
\inf_x \, a^T x + \|x \|_p = - \sup_x \, \langle - a, x \rangle - \| x \|_p = - f^*(-a).
\end{equation}
The conjugate of a norm is the indicator function for the dual norm unit ball.
Moreover, the dual norm for the $p$-norm is the $q$-norm, where $q = 1 - 1/p$. Thus,
$$
f^*(-a) = \begin{cases} 0 & \quad \text{if } \| a \|_q \leq 1, \\
\infty & \quad \text{otherwise.}
\end{cases}
$$
It follows that
$$
\inf_x \, a^T x + \|x \|_p  = \begin{cases} 0 & \quad \text{if } \| a \|_q \leq 1, \\
-\infty & \quad \text{otherwise.}
\end{cases}
$$
