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

This paper considers time-varying nonconvex optimization problems, utilized to model optimal operational trajectories of systems governed by possibly nonlinear physical or logical models. Algorithms for tracking a Karush-Kuhn-Tucker point are synthesized, based on a regularized primal-dual gradient method. In particular, the paper proposes a feedback-based primal-dual gradient algorithm, where analytical models for system state or constraints are replaced with actual measurements. When cost and constraint functions are twice continuously differentiable, conditions for the proposed algorithms to have bounded tracking error are derived, and a discussion of their practical implications is provided. Illustrative numerical simulations are presented for an application in power systems.

[1]  Andrey Bernstein,et al.  A Composable Method for Real-Time Control of Active Distribution Networks with Explicit Power Setpoints , 2014, ArXiv.

[2]  Na Li,et al.  Distributed Regularized Primal-Dual Method: Convergence Analysis and Trade-offs , 2016, 1609.08262.

[3]  R. Tyrrell Rockafellar,et al.  Augmented Lagrangians and Applications of the Proximal Point Algorithm in Convex Programming , 1976, Math. Oper. Res..

[4]  Gabriela Hug,et al.  Projected gradient descent on Riemannian manifolds with applications to online power system optimization , 2016, 2016 54th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[5]  R. Tyrrell Rockafellar,et al.  An Euler-Newton Continuation Method for Tracking Solution Trajectories of Parametric Variational Inequalities , 2013, SIAM J. Control. Optim..

[6]  Mingyi Hong,et al.  Decomposing Linearly Constrained Nonconvex Problems by a Proximal Primal Dual Approach: Algorithms, Convergence, and Applications , 2016, ArXiv.

[7]  Geert Leus,et al.  Double smoothing for time-varying distributed multiuser optimization , 2014, 2014 IEEE Global Conference on Signal and Information Processing (GlobalSIP).

[8]  G. Hug,et al.  Online optimization in closed loop on the power flow manifold , 2017, 2017 IEEE Manchester PowerTech.

[9]  Steven H. Low,et al.  An Online Gradient Algorithm for Optimal Power Flow on Radial Networks , 2016, IEEE Journal on Selected Areas in Communications.

[10]  J. Bank,et al.  Development of a High Resolution, Real Time, Distribution-Level Metering System and Associated Visualization, Modeling, and Data Analysis Functions , 2013 .

[11]  ASHISH CHERUKURI,et al.  Saddle-Point Dynamics: Conditions for Asymptotic Stability of Saddle Points , 2015, SIAM J. Control. Optim..

[12]  B. V. Dean,et al.  Studies in Linear and Non-Linear Programming. , 1959 .

[13]  Ruggero Carli,et al.  Distributed Reactive Power Feedback Control for Voltage Regulation and Loss Minimization , 2013, IEEE Transactions on Automatic Control.

[14]  Angelia Nedic,et al.  Subgradient Methods for Saddle-Point Problems , 2009, J. Optimization Theory and Applications.

[15]  Alejandro Ribeiro,et al.  Self-triggered time-varying convex optimization , 2016, 2016 IEEE 55th Conference on Decision and Control (CDC).

[16]  Lizhi Cheng,et al.  Precompact convergence of the nonconvex Primal-Dual Hybrid Gradient algorithm , 2018, J. Comput. Appl. Math..

[17]  Emiliano Dall'Anese,et al.  Dynamic ADMM for real-time optimal power flow , 2017, 2017 IEEE Global Conference on Signal and Information Processing (GlobalSIP).

[18]  Hao Jan Liu,et al.  Fast Local Voltage Control Under Limited Reactive Power: Optimality and Stability Analysis , 2015, IEEE Transactions on Power Systems.

[19]  Angelia Nedic,et al.  Multiuser Optimization: Distributed Algorithms and Error Analysis , 2011, SIAM J. Optim..

[20]  Wei Ren,et al.  Distributed Continuous-Time Convex Optimization With Time-Varying Cost Functions , 2017, IEEE Transactions on Automatic Control.

[21]  Emiliano Dall'Anese,et al.  Optimal power flow pursuit , 2016, 2016 American Control Conference (ACC).

[22]  Andrey Bernstein,et al.  A composable method for real-time control of active distribution networks with explicit power setpoints. Part I: Framework , 2015 .

[23]  Krishnamurthy Dvijotham,et al.  Real-Time Optimal Power Flow , 2017, IEEE Transactions on Smart Grid.

[24]  Andrey Bernstein,et al.  Explicit Conditions on Existence and Uniqueness of Load-Flow Solutions in Distribution Networks , 2016, IEEE Transactions on Smart Grid.

[25]  R. Rockafellar,et al.  Implicit Functions and Solution Mappings , 2009 .

[26]  Mingyi Hong,et al.  Gradient Primal-Dual Algorithm Converges to Second-Order Stationary Solutions for Nonconvex Distributed Optimization , 2018, ArXiv.

[27]  J. M. Martínez,et al.  On sequential optimality conditions for smooth constrained optimization , 2011 .

[28]  Enrique Mallada,et al.  High-Voltage Solution in Radial Power Networks: Existence, Properties, and Equivalent Algorithms , 2017, IEEE Control Systems Letters.

[29]  Yujie Tang,et al.  Distributed algorithm for time-varying optimal power flow , 2017, 2017 IEEE 56th Annual Conference on Decision and Control (CDC).