Please provide more details on the exact problem you wish to solve, and the properties (see below) of the weights.

As discussed in my post at Maximize the minimum singular value , the minimum singular value (for matrix dimension > 1) is neither convex nor concave. The minimum singular value is the special case of weighted nuclear norm having weight = 1 on the minimum singular value and weight = 0 on all other singular values. Therefore, the general weighted nuclear norm is not convex, and therefore can not be handled in CVX.

However, see the comments by Michael Grant, the CVX developer, at https://math.stackexchange.com/questions/1081446/how-is-the-nuclear-norm-convex-whereas-the-weighted-nuclear-norm-is-not , discussing the requirements on the weights in order for the weighted nuclear norm to be convex.

Also see sections 2.2.1, 2.2.2, 2.2.3 of “Weighted Nuclear Norm Minimization with Application to Image Denoisin” Gu, Zhang et al http://www4.comp.polyu.edu.hk/~cslzhang/paper/WNNM.pdf .relating to ordering of weights.