I was reading this paper where the define an optimization problem as
where K and L are kernel matrices and $\pi$ is the permutation matrix. They have explained that the function is convex because
I didn't get how they said convex non-decreasing square function. Where is the square function. The square is because of the Frobenius norm isn't it? and how come the function is non decreasing. Any clarification will be much appreciated.
A function $f(\mathbf{x})$ is convex if it can be expressed as the composition of a convex function $g$ with a 1-dimensional nondecreasing convex function $\phi$:
$$ f(\mathbf{x})=\phi(g(\mathbf{x})) $$
In this case, the convex function $g$ is a norm (the Frobenius norm), and the 1-dimensional nondecreasing convex function $\phi$ is the square function $\phi(z) = z^2$. I guess the authors decided to mangle their qualifiers with specifiers for sake of brevity.