Extension of the generalized distributed optimization algorithm: discrete time and discrete variables algorithm

Abstract

In this paper, we show that our previous generalized distributed optimization algorithms described by time differential equations can be realized in a form of time difference equations satisfying the global convergence to the optimal solutions by employing the notion of passivity. A key idea for the theory is on the appropriate transformation from the differential equations to the difference equations without loss of the passivity of the original algorithms as input/output systems. We show such a transformation and the corresponding theoretical development based on passivity is also possible for the time difference type algorithms. Furthermore, we also consider a case that some of the optimized variables are restricted to discrete numbers and discuss the stability or the convergence behavior of the solutions.