By adding a bilinear constraint in the form $x^T A y=0$, is this problem still convex-concave? I have a function $f(x,y)$ which is convex with respect to $x$ and concave to $y$. The problem 
\begin{align}
\min_x \max_y f(x,y)\\
\text{s.t.} x \in \mathcal{C}
\end{align} 
is then convex-concave saddle point. Now I found that I have to add another constraint in the form $x^\mathsf{T}A y=0$, where $A$ is an invertible matrix. Is the following problem still convex-concave?
\begin{align}
\min_x & \max_y f(x,y)\\
\text{s.t.} &x\in \mathcal{C}\\
&x^\mathsf{T} A y = 0
\end{align} 
I know that a problem of the form 
\begin{align}
\min_x & \max_y x^\mathsf{T} A y\\
\text{s.t.} &x\in \mathcal{C}\\
\end{align}
is convex-concave. 
 A: We have the bilinear equation
$$\rm x^{\top} A \, y = 0$$
where $\mathrm A \in \mathbb R^{n \times n}$ is a given invertible matrix. The equation above can be rewritten as follows
$$\begin{bmatrix} \mathrm x\\ \mathrm y\end{bmatrix}^{\top} \begin{bmatrix} \mathrm O_n & \mathrm A\\ \,\mathrm A^{\top} & \,\mathrm O_n\end{bmatrix} \begin{bmatrix} \mathrm x\\ \mathrm y\end{bmatrix} = 0$$
The characteristic polynomial of the block matrix is
$$\det \begin{bmatrix} s \mathrm I_n & -\mathrm A\\ -\mathrm A^{\top} & s\mathrm I_n\end{bmatrix} = \det \left( s^2 \mathrm I_n - \mathrm A \mathrm A^{\top} \right)$$
Since $\mathrm A$ is invertible, $\mathrm A \mathrm A^{\top}$ is positive definite. Thus, the spectrum of the block matrix contains the positive and negative square roots of the positive eigenvalues of $\mathrm A \mathrm A^{\top}$. Hence,
$$\begin{bmatrix} \mathrm O_n & \mathrm A\\ \,\mathrm A^{\top} & \,\mathrm O_n\end{bmatrix}$$
is indefinite. We then conclude that the set
$$\left\{ \begin{bmatrix} \mathrm x\\ \mathrm y\end{bmatrix} \in \mathbb R^{2n} : \begin{bmatrix} \mathrm x\\ \mathrm y\end{bmatrix}^{\top} \begin{bmatrix} \mathrm O_n & \mathrm A\\ \,\mathrm A^{\top} & \,\mathrm O_n\end{bmatrix} \begin{bmatrix} \mathrm x\\ \mathrm y\end{bmatrix} = 0 \right\}$$
is non-convex.
A: A standard approach can be used to convert your program into a single level optimization problem.  
Start with
$\min_{x} \max_{y} f(x,y)$
subject to 
$x \in C$
$x^{T}Ay=0$.
Now, assuming that the Karush-Kuhn Tucker sufficient conditions for the inner maximization are satisfied, we can substitute them into the problem.  For any given $x$, the KKT conditions for the inner maximization are:
$\nabla_{y} f(x,y)=\lambda A^{T}x $
$x^{T}Ay=0$.
The original problem can then be written as:
$\min_{x,y,\lambda} f(x,y)$
subject to
$x \in C$
$x^{T}Ay=0$
$\nabla_{y} f(x,y)=\lambda A^{T}x $.
Whether you can solve this will depend on the details of $f$ but it won't be a convex problem.  
