这是一份liverpool利物浦大学MATH367的成功案例

Let $X^{}$ be a solution to the optimization problem: Minimize $f(X)$ subject to: $$ X \in \mathcal{K} \text {, } $$ where $f$ is continuously differentiable and $\mathcal{K}$ is closed and convex. Then $X^{}$ is a solution of the variational inequality problem: determine $X^{} \in \mathcal{K}$, such that $$ \left\langle\nabla f\left(X^{}\right), X-X^{*}\right\rangle \geq 0, \quad \forall X \in \mathcal{K},
$$
where $\nabla f(X)$ is the gradient vector of $f$ with respect to $X$.

MATH367 COURSE NOTES ：

A Nash equilibrium is a strategy vector
$$
X^{}=\left(X_{1}^{}, \ldots, X_{m}^{}\right) \in \mathcal{K}, $$ such that $$ U_{i}\left(X_{i}^{}, \hat{X}{i}^{}\right) \geq U{i}\left(X_{i}, \hat{X}{i}^{}\right), \quad \forall X{i} \in \mathcal{K}^{i}, \forall i
$$
where $\hat{X}{i}^{}=\left(X{1}^{}, \ldots, X_{i-1}^{}, X_{i+1}^{}, \ldots, X_{m}^{*}\right)$.
In other words, under Nash equilibrium, no unilateral deviation in strategy by any single player is profitable for that player.