Multi-dimensional, convex and differentiable function

Question

I want to test an optimization procedure for multi-dimensional, convex and differentiable functions. Could anyone refer me to some popular functions that fit these criterions?

Answer 1

Remember that a matrix $H\in\mathbb{R}^{n\times n}$ is positive semi-definite if $y^THy\geq 0$ for all $y\in\mathbb{R}^n$, and positive definite if and only if $y^THy>0$ for all nonzero $y$.

Now consider the function $g_{x,y}(t)=f(x+ty)$, where $f$ is a function on $\mathbb{R}^n$ and $x,y$ are fixed vectors.

The derivatives at $t=0$ are $$g_{x,y}'(0) = y^T \nabla f(x), \qquad g_{x,y}''(0) = y^T \nabla^2 f(x) y$$ Of course if $g_{x,y}''(0)\geq 0$, then $g_{x,y}$ is convex at the origin. If $g_{x,y}''(0)\geq 0$ for all $y$, then $f$ is convex at $x$. Finally, if $g_{x,y}''(0)\geq 0$ for all $x,y$, then $f$ is convex.

These family of $g$'s can give one set of examples into the criteria you seek.