I wonder whether the following two optimization problems are equivalent. For matrices A and B, $A \preceq B$ means $B-A$ is positive semidefinite. "Tr" refers to the trace of matrix. $Z$ is an n-dimensional column vector, $V$ and $U$ are both square matrices.
Problem 1: \begin{equation} \begin{aligned} \max \quad & 1^{T}z-\frac{1}{n}\textbf{1}^{T}(V-U)\textbf{1}\\ \textrm{s.t.} \quad & 0 \preceq U,\hspace{0.1cm} 0 \preceq V, Tr(U+V)=1\\ & \frac{z_i+z_j}{2} \leq U_{i,j}-V_{i,j} \end{aligned} \end{equation}
Problem 2: $\lambda_i(Y)$ refers to the singular value of Y: \begin{equation} \begin{aligned} \max \quad & 1^{T}z\\ \textrm{s.t.} \quad & Y\textbf{1}=0,\sum_{i=1}^{n} |\lambda_i(Y)| \leq 1\\ & \frac{z_i+z_j}{2} \leq Y_{i,j} \end{aligned} \end{equation}
I can see if we let $Y=V-U$, then the problem would be very similar. But I don't see why $Y \textbf{1}=0$ specifically.
I want to add a bit of context why I think these two optimization problem are equivalent. In section 4.1 of this paper https://web.stanford.edu/~boyd/papers/pdf/fmmc.pdf, the author claims that problem 2 is the dual problem of finding the fastest mixing Markov chain on graph, which can be formulated into semidefinite programming (See section 2.3 equation 6). However, in Boyd and Vandenberghe's convex optimization book https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf, exercise 5.41, we are required to prove the dual problem can be formulated into problem 1. Therefore, I think these two problems must be equivalent. However, I can't spot the equivalence right now.