Analysis and Synthesis of Switched Optimization Algorithms

Deployment of optimization algorithms on networked systems face challenges associated with time delays and corruptions. One particular instance is the presence of time-varying delays arising from factors such as packet drops and irregular sampling. Fixed time delays can destabilize gradient descent algorithms, and this degradation is exacerbated by time-varying delays. This work concentrates on the analysis and creation of discrete-time optimization algorithms with certified exponential convergence rates that are robust against switched uncertainties between the optimizer and the gradient oracle. These optimization algorithms are implemented by a switch-scheduled output feedback controllers. Rate variation and sawtooth behavior (packet drops) in time-varying delays can be imposed through constraining switching sequences. Analysis is accomplished by bisection in the convergence rate to find Zames-Falb filter coefficents. Synthesis is performed by alternating between a filter coefficient search for a fixed controller, and a controller search for fixed multipliers.