Let $\Omega\subset \mathbb R^d$ be compact, $\rho$ be a density function on $\Omega$ and $p_1,\ldots, p_n\in (0,1)$ be weights satisfying $\int_{\Omega}\rho(z)dz=1=\sum_{k=1}^n p_k$. We consider the optimization problem $\inf_{x\in\Omega^n}\sup_{y\in\mathbb R^n}F(x,y)$, where $F$ is defined by

$$F(x,y)~:=~\sum_{k=1}^n\int_{V_k(x,y)}\left\{|z-x_k|^2-y_k\right\}\rho(z)dx+\sum_{k=1}^n p_ky_k,$$

with $x:=(x_1,\ldots, x_n)\in\Omega^n$, $y:=(y_1,\ldots, y_n)\in\mathbb R^n$ and

$$V_k(x,y)~:=~\big\{z\in\Omega:~ |z-x_k|^2-y_k\le |z-x_{i}|^2-y_{i},~ \forall 1\le i\le n\big\}.$$

My question is whether there exists any known computational scheme for solving numerically $\inf_{x\in\Omega^n}\sup_{y\in\mathbb R^n}F(x,y)$?

Some thoughts: This question appears in Reference request on Min-Max theorem where I asked $\inf_{x\in\Omega^n}\sup_{y\in\mathbb R^n}F(x,y)=\sup_{y\in\mathbb R^n}\inf_{x\in\Omega^n}F(x,y)$. To the best of my knowledge (*under suitable conditions*):

(1) the first order partial derivatives $\partial_x F$ and $\partial_y F$ have explicit expression;

(2) the second order partial derivatives $\partial_{xx} F$ and $\partial_{yy} F$ have explicit expression;

(3) For fixed $x\in\Omega^n$, $\sup_{y\in\mathbb R^n}F(x,y)$ can be numerically computed (related to optimal transport);

(4) For fixed $y\in\mathbb R^n$, $\inf_{x\in\Omega^n}F(x,y)$ can be numerically computed (related to Lloyd's algorithm).

Any comments or references are highly appreciated!