An Optimal Control Theory for Accelerated Optimization

Abstract

The first-order optimality conditions for a generic nonlinear optimization problem are generated as part of the terminal transversality conditions of an optimal control problem. It is shown that the Lagrangian of the optimization problem is connected to the Hamiltonian of the optimal control problem via a zero-Hamiltonian, infinite-order, singular arc. The necessary conditions for the singular optimal control problem are used to produce an auxiliary controllable dynamical system whose trajectories generate algorithm primitives for the optimization problem. A three-step iterative map for a generic algorithm is designed by a semi-discretization step. Neither the feedback control law nor the differential equation governing the algorithm need be derived explicitly. A search direction is produced by a proximal-aiming-type method that dissipates a control Lyapunov function. New step size procedures based on minimizing control Lyapunov functions along a search vector complete the design of the accelerated algorithms.