Control interpretations for first-order optimization methods

Hu, Bin; Lessard, Laurent

doi:10.23919/acc.2017.7963426

Cited by 37 publications

(38 citation statements)

References 19 publications

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“…The crux of our solution is a modularized architecture that breaks down the feedback dynamics (9) into three serial components that systematically addresses 2 Relaxing this assumption is desired and is a subject of future work. 3 The feedback interconnection of (7) and (9) is well-posed if u(t) and y(t) are uniquely defined for every choice of states x(t) and η(t).…”

Section: Optimization-based Control Designmentioning

confidence: 99%

“…For this analysis, it is useful to group the linear dynamics of E and D into the LTI system to essentially create a larger dimension LTI system. The resulting system iṡ Theorem 2: Consider the interconnection of the LTI system (7) and nonlinear feedback (13) with the equilibrium point considered in Theorem 1. Assume the pointwise IQC (Q, z * , ϕ(z * )) is satisfied and the assumptions of Theorem 1 hold.…”

Section: B Stability Analysismentioning

confidence: 99%

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An integral quadratic constraint framework for real-time steady-state optimization of linear time-invariant systems

Nelson

Mallada

2018

2018 Annual American Control Conference (ACC)

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Achieving optimal steady-state performance in real-time is an increasingly necessary requirement of many critical infrastructure systems. In pursuit of this goal, this paper builds a systematic design framework of feedback controllers for Linear Time-Invariant (LTI) systems that continuously track the optimal solution of some predefined optimization problem. The proposed solution can be logically divided into three components. The first component estimates the system state from the output measurements. The second component uses the estimated state and computes a drift direction based on an optimization algorithm. The third component computes an input to the LTI system that aims to drive the system toward the optimal steady-state.We analyze the equilibrium characteristics of the closed-loop system and provide conditions for optimality and stability. Our analysis shows that the proposed solution guarantees optimal steady-state performance, even in the presence of constant disturbances. Furthermore, by leveraging recent results on the analysis of optimization algorithms using integral quadratic constraints (IQCs), the proposed framework is able to translate input-output properties of our optimization component into sufficient conditions, based on linear matrix inequalities (LMIs), for global exponential asymptotic stability of the closed loop system. We illustrate the versatility of our framework using several examples.

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Section: Optimization-based Control Designmentioning

confidence: 99%

Section: B Stability Analysismentioning

confidence: 99%

An integral quadratic constraint framework for real-time steady-state optimization of linear time-invariant systems

Nelson

Mallada

2018

2018 Annual American Control Conference (ACC)

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“…Another interesting question is how to further improve the H 2 performance of saddle-point methods by designing auxiliary feedback controllers; augmentation is but one approach. Finally, extending these results beyond the case of quadratic cost functions with linear constraints will require nonlinear, robust control approaches, as in [21]- [25].…”

Section: I C O N C L U S I O N Smentioning

confidence: 99%

“…Indeed, an algorithm with a fast convergence rate would be inappropriate for control applications if it responded poorly to disturbances during transients, or if it greatly amplifies measurement noise in steady-state.The appropriate tool for measuring dynamic algorithm performance is instead the system norm, as commonly used in feedback system analysis to capture system response to exogenous disturbances. Recent work in this direction includes inputto-state-stability results [21], [22], finite L 2 -gain analysis [23], and the robust control framework proposed in [24], [25]. The purpose of this paper is to continue this line of investigation.…”

mentioning

confidence: 99%

Input–Output Performance of Linear–Quadratic Saddle-Point Algorithms With Application to Distributed Resource Allocation Problems

Simpson-Porco

Poolla

Monshizadeh

et al. 2020

IEEE Trans. Automat. Contr.

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Saddle-point or primal-dual methods have recently attracted renewed interest as a systematic technique to design distributed algorithms which solve convex optimization problems. When implemented online for streaming data or as dynamic feedback controllers, these algorithms become subject to disturbances and noise; convergence rates provide incomplete performance information, and quantifying input-output performance becomes more important. We analyze the input-output performance of the continuous-time saddle-point method applied to linearly constrained quadratic programs, providing explicit expressions for the saddle-point H2 norm under a relevant input-output configuration. We then proceed to derive analogous results for regularized and augmented versions of the saddlepoint algorithm. We observe some rather peculiar effectsa modest amount of regularization significantly improves the transient performance, while augmentation does not necessarily offer improvement. We then propose a distributed dual version of the algorithm which overcomes some of the performance limitations imposed by augmentation. Finally, we apply our results to a resource allocation problem to compare the inputoutput performance of various centralized and distributed saddlepoint implementations and show that distributed algorithms may perform as well as their centralized counterparts. I . I N T R O D U C T I O NSaddle-point methods are a class of continuous-time gradient-based algorithms for solving constrained convex optimization problems. Introduced in the early 1950s [1], [2], these algorithms are designed to seek the saddle points of the optimization problem's Lagrangian function. These saddle points are in one-to-one correspondence with the solutions of the first-order optimality (KKT) conditions, and the algorithm therefore drives its internal state towards the global optimizer of the convex program; see [3]-[5] for convergence results.Recently, these algorithms have attracted renewed attention for e.g., in the context of machine learning [6], in the control literature for solving distributed convex optimization problems [7], where agents cooperate through a communication network to solve an optimization problem with minimal or no centralized coordination. Applications of distributed optimization J. W. Simpson-Porco is with the include utility maximization [3], congestion management in communication networks [8], and control in power systems [9]- [14]. While most standard optimization algorithms require centralized information to compute the optimizer, saddle-point algorithms often yield distributed strategies in which agents perform state updates using only locally measured information and communication with some subset of other agents. We refer the reader to [4], [15]-[20] for control-theoretic interpretations of these algorithms.Rather than solve the optimization problem offline, it is desirable to run these distributed algorithms online as controllers, in feedback with system and/or disturbance measurements, to provide references so t...

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“…A framework based on integral quadratic constraints from robust control theory is proposed in [15] to analyze and design (centralized) iterative optimization algorithms. In [16] authors propose a loop-shaping interpretations for several existing optimization methods based on basic control elements such as PID and lag compensators. The convergence of distributed optimization algorithms by means of proper semidefinite programs is, instead, discussed in [17].…”

Section: Introductionmentioning

confidence: 99%

A System Theoretical Perspective to Gradient-Tracking Algorithms for Distributed Quadratic Optimization

Bin

Notarnicola

Marconi

et al. 2019

2019 IEEE 58th Conference on Decision and Control (CDC)

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other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. AbstractIn this paper we consider a recently developed distributed optimization algorithm based on gradient tracking. We propose a system theory framework to analyze its structural properties on a preliminary, quadratic optimization set-up.Specifically, we focus on a scenario in which agents in a static network want to cooperatively minimize the sum of quadratic cost functions. We show that the gradient tracking distributed algorithm for the investigated program can be viewed as a sparse closed-loop linear system in which the dynamic state-feedback controller includes consensus matrices and optimization (stepsize) parameters. The closed-loop system turns out to be not completely reachable and asymptotic stability can be shown restricted to a proper invariant set. Convergence to the global minimum, in turn, can be obtained only by means of a proper initialization. The proposed system interpretation of the distributed algorithm provides also additional insights on other structural properties and possible design choices that are discussed in the last part of the paper as a starting point for future developments.

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Control interpretations for first-order optimization methods

Cited by 37 publications

References 19 publications

An integral quadratic constraint framework for real-time steady-state optimization of linear time-invariant systems

An integral quadratic constraint framework for real-time steady-state optimization of linear time-invariant systems

Input–Output Performance of Linear–Quadratic Saddle-Point Algorithms With Application to Distributed Resource Allocation Problems

A System Theoretical Perspective to Gradient-Tracking Algorithms for Distributed Quadratic Optimization

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