This rule is sometimes called the <em>partial
minimization rule</em>.) We can think of the convex function <span class="math">\(f\)</span> as
the optimal value of a family of convex optimization problems, indexed
or parametrized by <span class="math">\(x\)</span>,</p>
<div class="math">
\[\begin{split}\begin{array}{ll}
    \mbox{minimize} &amp; g(x,y) \\
    \mbox{subject to} &amp; (x,y) \in S
\end{array}\end{split}\]</div>
<p>with optimization variable <span class="math">\(y\)</span>.</p>
<p>One special ca