Convex functional
= Convex functional
A <functional> on a real <vector space> is convex when it is a <convex function>: $J(\theta v+(1-\theta)w)\leq\theta J(v)+(1-\theta)J(w)$ for $0\leq\theta\leq1$. For a symmetric nonnegative <bilinear form> $a$ and a <linear functional> $\ell$, the quadratic functional $J(v)=\tfrac12a(v,v)-\ell(v)$ is convex, since its convexity difference is $\tfrac12\theta(1-\theta)a(v-w,v-w)\geq0$.