@MarkL.Stone Strict convexity can be used to prove my conclusion. However, my conclusion is possible to be true for convex functions which are not strictly convex.

@MarkL.Stone I plotted the objective values in the case of two dimensions. I found what I want to prove is true. And I believe it's also true for higher dimensions. I think the main difficulty is from the unusual definition of nuclear norm of a matrix. The definition seems disconnected from the original elements of the matrix because of the operation of SVD. So I can't find the proper tool to study the behavior of nuclear norm with respect to the change of elements of the matrix.

@MarkL.Stone My question is to prove that the optimal solution is only attained at the extreme point of $S$ and can't be other points. That is to say, being an extreme point is a necessary condition for an optimal solution. I gave a counterexample based on losif Pinelis's answer. For example, $F(X)=1$ is a convex function, but the optimal solution is attained at any feasible solution. Thus, it's not enough to only depend on the convexity of nuclear norm.

I'm sorry but there is ambiguity in my question. I mean the optimal solution in only attained at extreme points. I have updated the question accordingly. For example, $F(X)=1$ is a convex function, but the optimal solution is attained at any feasible solution.

Thanks a lot for your answer. I can see it’s also a great idea.Though I tired hard to understand your proof, given the fact that my major is not mathematics, I can’t figure it out. I really appreciate your brilliant idea you put here.

Thank you so much for your excellent answer to the newest updated problem. Though I haven’t gone through the whole derivation yet because my limited mathematic knowledge, I verified the numerical equality between $F(n)$ and the proposed integral form using Matlab when p1, p2, and p3 were set to some specific values. I believe your answer is correct and I really appreciate it because it helps me a lot. I will keep working on it till I fully understand it.

A perfect answer! Thank you very much.