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1. A distributed continuous-time modified Newton–Raphson algorithm

   https://doi.org/https://doi.org/10.1016/j.automatica.2021.109886  

   Moradian, Hossein; Kia, Solmaz S. (January 2022, Automatica)

   We propose a continuous-time second-order optimization algorithm for solving unconstrained convex optimization problems with bounded Hessian. We show that this alternative algorithm has a comparable convergence rate to that of the continuous-time Newton–Raphson method, however structurally, it is amenable to a more efficient distributed implementation. We present a distributed implementation of our proposed optimization algorithm and prove its convergence via Lyapunov analysis. A numerical example demonstrates our results. 

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2. On Matrix Momentum Stochastic Approximation and Applications to Q-learning

   https://doi.org/10.1109/ALLERTON.2019.8919828  

   Devraj, Adithya M.; Busic, Ana; Meyn, Sean (September 2019, Allerton Conference on Communication, Control, and Computing)

   null (Ed.)

   Stochastic approximation (SA) algorithms are recursive techniques used to obtain the roots of functions that can be expressed as expectations of a noisy parameterized family of functions. In this paper two new SA algorithms are introduced: 1) PolSA, an extension of Polyak's momentum technique with a specially designed matrix momentum, and 2) NeSA, which can either be regarded as a variant of Nesterov's acceleration method, or a simplification of PolSA. The rates of convergence of SA algorithms is well understood. Under special conditions, the mean square error of the parameter estimates is bounded by σ 2 /n+o(1/n), where σ 2 ≥ 0 is an identifiable constant. If these conditions fail, the rate is typically sub-linear. There are two well known SA algorithms that ensure a linear rate, with minimal value of variance, σ 2 : the Ruppert-Polyak averaging technique, and the stochastic Newton-Raphson (SNR) algorithm. It is demonstrated here that under mild technical assumptions, the PolSA algorithm also achieves this optimality criteria. This result is established via novel coupling arguments: It is shown that the parameter estimates obtained from the PolSA algorithm couple with those of the optimal variance (but computationally more expensive) SNR algorithm, at a rate O(1/n 2 ). The newly proposed algorithms are extended to a reinforcement learning setting to obtain new Q-learning algorithms, and numerical results confirm the coupling of PolSA and SNR. 

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3. Online Learning Guided Curvature Approximation: A Quasi-Newton Method with Global Non-Asymptotic Superlinear Convergence

   Jiang, Ruichen; Jin, Qiujiang; Mokhtari, Aryan (July 2023, PMLR)

   Quasi-Newton algorithms are among the most popular iterative methods for solving unconstrained minimization problems, largely due to their favorable superlinear convergence property. However, existing results for these algorithms are limited as they provide either (i) a global convergence guarantee with an asymptotic superlinear convergence rate, or (ii) a local non-asymptotic superlinear rate for the case that the initial point and the initial Hessian approximation are chosen properly. In particular, no current analysis for quasi-Newton methods guarantees global convergence with an explicit superlinear convergence rate. In this paper, we close this gap and present the first globally convergent quasi-Newton method with an explicit non asymptotic superlinear convergence rate. Unlike classical quasi-Newton methods, we build our algorithm upon the hybrid proximal extragradient method and propose a novel online learning framework for updating the Hessian approximation matrices. Specifically, guided by the convergence analysis, we formulate the Hessian approximation update as an online convex optimization problem in the space of matrices, and we relate the bounded regret of the online problem to the superlinear convergence of our method. 

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4. A Machine Learning Initializer for Newton-Raphson AC Power Flow Convergence

   https://doi.org/10.1109/TPEC60005.2024.10472261  

   Okhuegbe, Samuel N; Ademola, Adedasola A; Liu, Yilu (March 2024, 2024 IEEE Texas Power and Energy Conference (TPEC))

   Power flow computations are fundamental to many power system studies. Obtaining a converged power flow case is not a trivial task especially in large power grids due to the non-linear nature of the power flow equations. One key challenge is that the widely used Newton based power flow methods are sensitive to the initial voltage magnitude and angle estimates, and a bad initial estimate would lead to non-convergence. This paper addresses this challenge by developing a random-forest (RF) machine learning model to provide better initial voltage magnitude and angle estimates towards achieving power flow convergence. This method was implemented on a real ERCOT 6102 bus system under various operating conditions. By providing better Newton-Raphson initialization, the RF model precipitated the solution of 2,106 cases out of 3,899 non-converging dispatches. These cases could not be solved from flat start or by initialization with the voltage solution of a reference case. Results obtained from the RF initializer performed better when compared with DC power flow initialization, Linear regression, and Decision Trees. 

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5. Efficient Load Flow Techniques Based on Holomorphic Embedding for Distribution Networks

   https://doi.org/10.1109/PESGM40551.2019.8973882  

   Heidarifar, Majid; Andrianesis, Panagiotis; Caramanis, Michael (August 2019, IEEE PES General meeting, August 4-8 2019, Atlanta, GA, USA, 5 pages, 2019)

   null (Ed.)

   The Holomorphic Embedding Load flow Method (HELM) employs complex analysis to solve the load flow problem. It guarantees finding the correct solution when it exists, and identifying when a solution does not exist. The method, however, is usually computationally less efficient than the traditional Newton-Raphson algorithm, which is generally considered to be a slow method in distribution networks. In this paper, we present two HELM modifications that exploit the radial and weakly meshed topology of distribution networks and significantly reduce computation time relative to the original HELM implementation. We also present comparisons with several popular load flow algorithms applied to various test distribution networks. 

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