Fixed-Time Stable Gradient Flows: Applications to Continuous-Time Optimization

Garg, Kunal; Panagou, Dimitra

doi:10.1109/tac.2020.3001436

Cited by 69 publications

(33 citation statements)

References 32 publications

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“…We mentioned the extension to Riemannian manifolds, which should be quite straightforward. More interestingly, finite-and fixed-time gradient flows [5,26] may also be studied on open subsets and, most importantly in the current context, from the point of view of finite and fixed-time ISS in the sense of e.g. [10,21,9].…”

Section: Discussionmentioning

confidence: 99%

Remarks on input to state stability of perturbed gradient flows, motivated by model-free feedback control learning

Sontag¹

2021

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Recent work on data-driven control and reinforcement learning has renewed interest in a relative old field in control theory: model-free optimal control approaches which work directly with a cost function and do not rely upon perfect knowledge of a system model. Instead, an "oracle" returns an estimate of the cost associated to, for example, a proposed linear feedback law to solve a linear-quadratic regulator problem. This estimate, and an estimate of the gradient of the cost, might be obtained by performing experiments on the physical system being controlled. This motivates in turn the analysis of steepest descent algorithms and their associated gradient differential equations. This note studies the effect of errors in the estimation of the gradient, framed in the language of input to state stability, where the input represents a perturbation from the true gradient. Since one needs to study systems evolving on proper open subsets of Euclidean space, a self-contained review of input to state stability definitions and theorems for systems that evolve on such sets is included. The results are then applied to the study of noisy gradient systems, as well as the associated steepest descent algorithms.

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Section: Discussionmentioning

confidence: 99%

Remarks on input to state stability of perturbed gradient flows, motivated by model-free feedback control learning

Sontag¹

2021

Preprint

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“…In [27], the authors design a modified gradient flow scheme with fixed-time convergence guarantees assuming that the objective function is strongly convex. In [17], modified gradient flow schemes are introduced for unconstrained and constrained convex optimization problems, as well as for min-max problems posed as convex-concave optimization problems. The work in [17] only considered linear equality constraints, and assumed that the objective function is continuously differentiable, and satisfies strong or strict convexity, or is gradient-dominated.…”

Section: Mathematics Subject Classification (2000) 34d20 • 37n40 • 47...mentioning

confidence: 99%

“…In [17], modified gradient flow schemes are introduced for unconstrained and constrained convex optimization problems, as well as for min-max problems posed as convex-concave optimization problems. The work in [17] only considered linear equality constraints, and assumed that the objective function is continuously differentiable, and satisfies strong or strict convexity, or is gradient-dominated. The schemes proposed in this paper apply to a broader class of problems, namely, MVIPs, and non-smooth convex optimization problems arise as special cases of the general framework considered in this paper.…”

Section: Mathematics Subject Classification (2000) 34d20 • 37n40 • 47...mentioning

confidence: 99%

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Fixed-Time Stable Proximal Dynamical System for Solving MVIPs

Garg,

Baranwal,

Gupta

et al. 2019

Preprint

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In this paper, the fixed-time stability of a novel proximal dynamical system is investigated for solving mixed variational inequality problems. Under the assumptions of strong monotonicity and Lipschitz continuity, it is shown that the solution of the proposed proximal dynamical system exists in the classical sense, is uniquely determined and converges to the unique solution of the associated mixed variational inequality problem in a fixed time. As a special case, the proposed proximal dynamical system reduces to a novel fixed-time stable projected dynamical system. Furthermore, the fixed-time stability of the modified projected dynamical system continues to hold, even if the assumptions of strong monotonicity are relaxed to that of strong pseudomonotonicity. Connections to convex optimization problems are discussed, and commonly studied dynamical systems in the continuous-time optimization literature are shown as special cases of the proposed proximal dynamical system considered in this paper. Finally, several numerical examples are presented that corroborate the fixed-time convergent behavior of the proposed proximal dynamical system.

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“…The ability to reach the optimum of a cost function of interest in a prescribed time can provide added flexibility in engineering design problems requiring precise synchronous control tasks. In continuous-time systems, gradient-based descent algorithms have been proposed 1,2,3 . In 2,3 1 a class of discontinuous scaled gradient/Hessian dynamical systems was proposed for the design of continuous-time optimization algorithms with finite-time convergence.…”

Section: Introductionmentioning

confidence: 99%

Finite-time Newton seeking control

Guay,

Benosman

2020

Preprint

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This paper proposes a finite-time Newton seeking control design for systems described by unknown multivariable static maps. The Newton seeking system has an averaged system that implements a Newton continuous-time algorithm. The averaged Newton seeking system is shown to achieve finite-time stability of the unknown optimum of the static map. An averaging analysis demonstrates that the proposed Newton-seeking achieves finite-time practical stability of the optimum. A simulation study is used to demonstrate the effectiveness of the design method.

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Fixed-Time Stable Gradient Flows: Applications to Continuous-Time Optimization

Cited by 69 publications

References 32 publications

Remarks on input to state stability of perturbed gradient flows, motivated by model-free feedback control learning

Remarks on input to state stability of perturbed gradient flows, motivated by model-free feedback control learning

Fixed-Time Stable Proximal Dynamical System for Solving MVIPs

Finite-time Newton seeking control

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