Characterizations of Input-to-State Stability in Nonlinear Optimization Algorithms

Abstract: Nonlinear optimization has become increasingly involved in the guidance and control of dynamic systems. Its applications include optimal path planning, collision avoidance, model predictive control, and extremum seeking. This has motivated the development of an algorithmic systems theory, which studies the stability and robustness of optimization algorithms as dynamic systems. This talk focuses on the relationship between input-to-state stability of nonlinear optimization under perturbation and (strong) metric regularity of the Karush—Kuhn—Tucker system of necessary conditions. Rooted in variational analysis, metric regularity is a notion of Lipschitz stability for a primal-dual solution under perturbations. It has played a prominent role in analyzing Newton-type methods for optimization, including sequential quadratic programming and augmented Lagrangian methods. In my talk, I characterize strong metric regularity by necessary and sufficient optimality conditions. Moreover, I show that strong metric regularity implies, but is stronger than, small-input input-to-state stability of a prototypical Newton method. These results show that metric regularity plays a significant role in the systems theory of nonlinear optimization algorithms.

Biography: Torbjørn Cunis received his doctoral degree in systems and control from ISAE-Supaéro, University of Toulouse, in 2019. Before that, he studied computer science, aerospace computer engineering, and automation engineering at the University of Würzburg and RWTH Aachen University. Since 2021, he has been a lecturer (Akademischer Rat a.Z.) at the University of Stuttgart Institute of Flight Mechanics and Controls and an adjunct researcher at the University of Michigan Aerospace Department. He was a researcher at ONERA – The French Aerospace Lab (with Laurent Burlion) from 2016 to 2019 and a research fellow at the University of Michigan (with Ilya Kolmanovsky) from 2019 to 2021. His research focuses on algorithmic systems and control theory, in particular, nonlinear optimization algorithms and verifiable nonlinear control systems. Dr. Cunis is a fellow of the Young ZiF at the Centre for Interdisciplinary Research at the University of Bielefeld.

References:
[1] T. Cunis and I. Kolmanovsky, ‘Input-to-State Stability of Newton Methods for Generalized Equations in Nonlinear Optimization’, in 2024 IEEE Conference on Decision and Control, Milano, 2024, pp. 5950–5956. doi: 10.1109/CDC56724.2024.10885904.
[2] T. Cunis and I. Kolmanovsky, ‘Inexactness in Bilevel Nonlinear Optimization: A Gradient-free Newton’s Method Approach’, in Symposium on Systems Theory in Data and Optimization, Stuttgart, 2024.
[3] T. Cunis and I. Kolmanovsky, ‘Input-to-State Stability of a Bilevel Proximal Gradient Descent Algorithm’, in 22nd IFAC World Congress, Yokohama, 2023, pp. 7474–7479. doi: 10.1016/j.ifacol.2023.10.633.

systems and controlanalysis of PDEsclassical analysis and ODEsdynamical systemsoptimization and control

Audience: researchers in the topic

( slides | video )

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Input-to-State Stability and its Applications

Series comments: This is a seminar for the exchange of ideas in input-to-state stability (ISS) theory and related fields.

The scope of the Seminar includes but is not limited to

- ISS for finite-dimensional systems (ODEs, hybrid, impulsive, switched, discrete-time systems),

- Infinite-dimensional ISS theory (PDEs, evolution equations in Banach spaces, time-delay systems, infinite networks)

- Applications to robust control and observation, nonlinear control, network analysis, etc.

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