Linear optimization problem of $argmin$
In response to the comments suggesting subgradient descent methods: the problem in the question does not contain any sum or similar terms, therefore the variables are independent of each other and there is a closed-form solution:
$$ x^* = 
\begin{cases}
    a - \frac{\lambda}{2},&amp; \text{if } a &gt; \frac{\lambda}{2}\\
    a + \frac{\lambda}{2},&amp; \text{if } -a &gt; \frac{\lambda}{2}\\
    0,              &amp; \text{otherwise}
\end{cases}
$$