nash01 单纯形 INERTIAL GAME DYNAMICS “heavy ball with friction” method.

几何学上，单纯形或者n-单纯形是和三角形类似的n维几何体。精确的讲，单纯形是某个n维以上的欧几里得空间中的（n+1）个仿射无关（也就是没有m维平面包含m+1个点；这样的点集被称为处于一般位置）的点的集合的凸包

INERTIAL GAME DYNAMICS AND

APPLICATIONS TO CONSTRAINED OPTIMIZATION

  

RIDA LARAKI AND PANAYOTIS MERTIKOPOULOS

Abstract. We derive a class of inertial dynamics for games and constrained optimization

problems over simplices by building on the well-known “heavy ball

with friction” method. In the single-agent case, the dynamics are generated by

endowing the problem’s configuration space with a Hessian–Riemannian structure

and then deriving the equations of motion for a particle moving under the

influence of the problem’s objective (viewed as a potential field); for normal form

 

games, the procedure is similar, but players are instead driven by the unilateral

gradient of their payoff functions. By specifying an explicit Nash–Kuiper embedding

of the simplex, we show that global solutions exist if and only if the interior

of the simplex is mapped isometrically to a closed hypersurface of some ambient

Euclidean space, and we characterize those Hessian–Riemannian structures

which have this property. For such structures, low-energy solutions are attracted

to isolated minimizers of the potential, showing in particular that pure Nash

equilibria of potential games are attracting; more generally, when the game is not

a potential one, we establish an inertial variant of the folk theorem of evolutionary

game theory,