Let's first describe the setup: we consider a(say smooth enough) function $f: \mathbb{R}^d \to \mathbb{R}$ and write it as $(x,y) \to f(x,y)$, where $x \in \mathbb{R}^{d_x}$, $y \in \mathbb{R}^{d_y}$ and $d=d_x+d_y$. The Hessian of $f$ can then be written as $$ H=\left( \begin{array}{rr} D_{xx}^2 f(x,y) & D_{yx}^2 f(x,y) \\ D_{xy}^2 f(x,y) & D_{yy}^2 f(x,y) \\ \end{array}\right). $$ On the other hand we consider the gradient dynamics of a game between $x$ and $y$, where $x$ wants to minimize, and $y$ to maximize $f$. If we write $$ w(z)=\left( \begin{array}{rr} -D_x f(x,y) \\ D_y f(x,y) \end{array}\right) ,$$ the corresponding differential equation can be written as $\dot{z}=w(z)$. The flow in the neighborhoods of equilibria can then be analyzed by examining the Jacobian of $w$: $$J=\left( \begin{array}{rr} -D_{xx}^2 f(x,y) & -D_{yx}^2 f(x,y) \\ D_{xy}^2 f(x,y) & D_{yy}^2 f(x,y) \\ \end{array}\right). $$ My question is now the following: Is it possible, that a critical point of $f$ is on the one hand a local minimum of $f$, but simultaneously a locally asymptotically stable equilibrium of the gradient dynamics from above. Or in other words: Is it possible, that simultaneously $H$ is positive definite and all eigenvalues of $J$ have negative real parts at a critical point of $f$? I know that positive definiteness of $H$ is equivalent to all principal minors of $H$ being positive, and the stability of $J$ can be analyzed by the Routh-Hurwitz criterion, unfortunately, I couldn't find a (probably very easy) argument or explicit example yet to answer the question. Any help is highly appreciated!


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