This formula is :
$$f(W,V,B) =\|XW-V\|^2_F +\|Y-VB\|^2_F +\operatorname{tr}(V'LV) +2\operatorname{tr}(W'DW),$$
where $X$, $Y$ are constant matrices and $L$ is constant laplace matrix. Suppose $D$ is a constant diagonal matrix.
After @ A.Γ.'s suggestion, I modified the question. The original question is as follows:
Is the convexity of a function with a saddle point multivariate function plus a multivariate convex function necessarily non-convex?