Convexity Preserved Under Monotone Convex Composition

Convexity Under Monotone Convex Composition: Let $X$ be a vector space . Suppose $f:X\to\overline{\mathbb{R}}$ is convex and $\phi:\mathbb{R}\to\overline{\mathbb{R}}$ is convex and nondecreasing on a convex set containing the range of $f$. Then the composition $\phi\circ f$ is convex on $X$.

This rule is a standard way to build convex penalties (e.g., $\phi(t)=e^t$ or $\phi(t)=t_+ := \max\{t,0\}$) from an existing convex function $f$.

Proof sketch (idea): Apply Jensen to $f$ to get $f(\lambda x+(1-\lambda)y)\le \lambda f(x)+(1-\lambda)f(y)$. Then use monotonicity of $\phi$ followed by Jensen for $\phi$.