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.

翻译：本研究提出了一种在连续时间域中设计加速凸优化算法的构造性方法。两个关键使能因素是控制理论中的经典无源性概念以及将内部动态系统输出映射到优化变量的时变变量变换。通过将驱动状态演化的内部动力学指定为无源线性时不变系统，优化动力学关联的李雅普诺夫函数自然得以确定。该无源性方法论提供了一个通用框架，可灵活生成能保证目标函数值具有不同收敛速率界的凸优化算法。通过重新定义内部动力学输出以允许与跟踪误差动力学进行反馈互联，同一原理也适用于自适应控制的在线参数更新算法设计。