A Feedback-Based Regularized Primal-Dual Gradient Method for Time-Varying Nonconvex Optimization

Abstract

This paper considers a nonconvex optimization problem that evolves over time, and addresses the synthesis and analysis of regularized primal-dual gradient methods to track a Karush-Kuhn-Tucker (KKT) trajectory. The proposed regularized primal-dual gradient methods are implemented in a running fashion, in the sense that the underlying optimization problem changes during the iterations of the algorithms. For a problem with twice continuously differentiable cost and constraints, and under a generalization of the Mangasarian-Fromovitz constraint qualification, sufficient conditions are derived for the running algorithm to track a KKT trajectory. Further, asymptotic bounds for the tracking error (as a function of the time-variability of a KKT trajectory) are obtained. A continuous-time version of the algorithm, framed as a system of differential inclusions, is also considered and analytical convergence results are derived. For the continuous-time setting, a set of sufficient conditions for the KKT trajectories not to bifurcate or merge is proposed. Illustrative numerical results inspired by a real-world application are provided.

Time-Varying Convex Optimization: Time-Structured Algorithms and Applications

“…Examples of works include real-time algorithms for voltage control, OPF, as well as DER management for aggregators (see, for example, [60]- [65] and pertinent references therein). For some applications, such as the demand response and the OPF, online algorithms have been designed to leverage measurements of constraints (e.g., voltages violations) in the algorithmic updates [18], [61], [66] to relax the sensing requirements. Real-time measurements were used in a state estimation framework in [67].…”

“…The algorithm produces decisions on setpoints for the active and reactive power outputs xi(t k ) of the DERs, which are commanded to the devices. Finally, "feedback" can come in the form of measurements of the actual power outputs xi(t k ) [60], as well as other electrical quantities [61], [66].…”

Pursuit of Low-Rank Models of Time-Varying Matrices Robust to Sparse and Measurement Noise

“…Second, we do not make restrictive assumptions on RIP properties, incoherence, identical covariance matrices, independence of all outlier supports, or initialisation. Broadly speaking, such analyses of time-varying non-convex optimisation (Liu et al 2018;Tang et al 2018;Fattahi et al 2019;Massicot and Marecek 2019), seems to be an important direction for further research.…”

The Optimal Power Flow Operator: Theory and Computation

“…In recent works we showed that "worst-case" sensitivity bounds provide privacy guarantees when releasing power flow data [16] and for data disaggregation [23]. In the context of real-time optimization where sensitivity is often assumed to be known and bounded [24], this work can be used to provide exactly these bounds (or rule them out).…”