where \mu_1 \geq \mu_2 \cdots \geq \mu_K. The optimization variables include \mathbf{d} = [d_1, \ldots, d_K] ^ T and \mathbf{Q}. The function is concave to \mathbf{d} and is convex to \mathbf{Q}.

is convex to \mathbf{Q}, so the function is convex to \mathbf{Q}.
To show that the function is concave to \mathbf{d}, we can use the following equation

You need the maximum (objective value) with respect to d to be convex with respect to Q. Otherwise, it is not a convex optimization problem.I.e., you need the objective function of the outer minimization to be jointly convex in d and Q.

Thanks for your reply. For now I can’t prove the convexity of the outer minimization.
Perhaps I need to implement a primal-dual algorithm myself to solve it.

As you mentioned that CVX can be used to solve some, I am interested in what kinds of minimax problems can be solved with CVX.

Ok. Now I know that CVX cannot be used to solve the minimax or saddle point problem of a continous convex-concave function f(x,y), unless the problem can be converted to a standard convex (or conic) problem.