A Passivity-Based Method for Accelerated Convex Optimisation
Namhoon Cho Hyo-Sang Shin
This study presents a constructive methodology for designing accelerated
convex optimisation algorithms in continuous-time domain. The two key enablers
are the classical concept of passivity in control theory and the time-dependent
change of variables that maps the output of the internal dynamic system to the
optimisation variables. The Lyapunov function associated with the optimisation
dynamics is obtained as a natural consequence of specifying the internal
dynamics that drives the state evolution as a passive linear time-invariant
system. The passivity-based methodology provides a general framework that has
the flexibility to generate convex optimisation algorithms with the guarantee
of different convergence rate bounds on the objective function value. The same
principle applies to the design of online parameter update algorithms for
adaptive control by re-defining the output of internal dynamics to allow for
the feedback interconnection with tracking error dynamics.