Recurrent network expression and its property of replicator dynamics for optimization

Replicator dynamics (RD) is a well-known mathematical model of evolutionary dynamics. In the study of optimization, a gradient dynamics called the variable metric gradient projection (VMGP) model, which is used to solve a constrained optimization problem with normalized equality and nonnegative inequalities, is known to have the structure of RD. In this paper, we show that the VMGP dynamics can also be considered to have the structure of recurrent neural network (N.N.) by introducing a new variable so as to transform the VMGP dynamics equivalently. We found that it is described as a new model similar to the well known Hopfield\´s N.N. by regarding the newly introduced variable as "inner state" and giving a particular nonlinear element as output unit of the network. We also provide some interesting properties of the network model through fixed point analysis for the nonlinear dynamics. Numerical simulations show the validity of our discussions.