A simplex particle swarm optimization (simplex-PSO) derived from the Nelder-Mead simplex method was proposed to optimize the high dimensionality functions. In simplex-PSO, the velocity term was abandoned and its reference objectives were the best particle and the centroid of all particles except the best particle. The convergence theorems of linear time-varying discrete system proved that simplex-PSO is of consistent asymptotic convergence. In order to reduce the probability of trapping into a local optimal value, an extremum mutation was introduced into simplex-PSO and simplex-PSO-t (simplex-PSO with turbulence) was devised. Several experiments were carried out to verify the validity of simplex-PSO and simplex-PSO-t, and the experimental results confirmed the conclusions: (1) simplex-PSO-t can optimize high-dimension functions with 200-dimensionality; (2) compared PSO with chaos PSO (CPSO), the best optimum index increases by a factor of 1×10~2-1×10~4.