r defining new functions in CVX relies on
the following basic result of convex analysis. Suppose that
<span class="math">\(S\subset\mathbf{R}^n\times\mathbf{R}^m\)</span> is a convex set and
<span class="math">\(g:(\mathbf{R}^n\times\mathbf{R}^m)\rightarrow(\mathbf{R}\cup+\infty)\)</span>
is a convex function. Then</p>
<div class="math">
\[f:\mathbf{R}^n\rightarrow(\mathbf{R}\cup+\infty), \quad f(x) \triangleq \inf\left\{\,g(x,y)\,~|~\,\exists y,~(x,y)\in S \,\right\}\]</div>
<p>is also a convex function. (