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Many modern large-scale and distributed optimization problems can be cast into a form in which the objective function is a sum of a smooth term and a nonsmooth regularizer. Such problems can be solved via a proximal gradient method which generalizes standard gradient descent to a nonsmooth setup. In this paper, we leverage the tools from control theory to study global convergence of proximal gradient flow algorithms. We utilize the fact that the proximal gradient algorithm can be interpreted as a variable-metric gradient method on the forward-backward envelope. This continuously differentiable function can be obtained from the augmented Lagrangian associated with the original nonsmooth problem and it enjoys a number of favorable properties. We prove that global exponential convergence can be achieved even in the absence of strong convexity. Moreover, for in-network optimization problems, we provide a distributed implementation of the gradient flow dynamics based on the proximal augmented Lagrangian and prove global exponential stability for strongly convex problems.
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Proximal algorithms for large-scale statistical modeling and sensor/actuator selection
Several problems in modeling and control of stochastically-driven dynamical systems can be cast as regularized semi-definite programs. We examine two such representative problems and show that they can be formulated in a similar manner. The first, in statistical modeling, seeks to reconcile observed statistics by suitably and minimally perturbing prior dynamics. The second seeks to optimally select a subset of available sensors and actuators for control purposes. To address modeling and control of large-scale systems we develop a unified algorithmic framework using proximal methods. Our customized algorithms exploit problem structure and allow handling statistical modeling, as well as sensor and actuator selection, for substantially larger scales than what is amenable to current general-purpose solvers. We establish linear convergence of the proximal gradient algorithm, draw contrast between the proposed proximal algorithms and alternating direction method of multipliers, and provide examples that illustrate the merits and effectiveness of our framework.
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- Award ID(s):
- 1809833
- PAR ID:
- 10128658
- Date Published:
- Journal Name:
- IEEE Transactions on Automatic Control
- ISSN:
- 0018-9286
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1109/TAC.2019.2948268
