arXiv subjects

Carsten Scherer

Publications and source records attributed to Carsten Scherer.

6 recordsLinked to original sources

arXiv2026

Structure, Analysis, and Synthesis of First-Order Algorithms

Optimization algorithms can be interpreted through the lens of dynamical systems as the interconnection of linear systems and a set of subgradient nonlinearities. This dynamical systems formulation allows for the analysis and synthesis of optimization algorithms by solving robust control problems. In this work, we use the celebrated internal model principle in control theory to structurally factorize convergent composite optimization algorithms into suitable network-dependent internal models and core subcontrollers. As the key benefit, we reveal that this permits us to synthesize optimization algorithms even if information is transmitted over networks featuring dynamical phenomena such as time delays, channel memory, or crosstalk. Design of these algorithms is achieved under bisection in the exponential convergence rate either through a nonconvex local search or by alternation of convex semidefinite programs. We demonstrate factorization of existing optimization algorithms and the automated synthesis of new optimization algorithms in the networked setting.

math.OC↗

arXiv2026

Global Stability and Step Size Robustness of RMSProp

In this paper, an input-to-state Lyapunov function for the RMSProp optimization algorithm is introduced. Global asymptotic stability of the RMSProp algorithm for constant step sizes and robustness properties with respect to arbitrary bounded time-varying step size rules are established.

math.OC↗

arXiv2025

Analysis and Synthesis of Switched Optimization Algorithms

Deployment of optimization algorithms over communication networks face challenges associated with time delays and corruptions. Fixed time delays can destabilize popular gradient-based algorithms, and this degradation is exacerbated by time-varying delays that may arise from packet drops. This work concentrates on the analysis and synthesis of discrete-time optimization algorithms with certified exponential convergence rates that are robust against switched network dynamics between the optimizer and the gradient oracle. Analysis is accomplished by solving linear matrix inequalities under bisection in the exponential convergence rate, searching over Zames-Falb filter coefficients that can certify convergence. Synthesis is performed by alternating between a search over filter coefficient for a fixed controller, and a search over controllers for a fixed filter. Effectiveness is demonstrated by the synthesis of convergent optimization algorithms over networks with time-varying delays, and networks with unstable channel dynamics.

math.OC↗

arXiv2021

Dissipativity and Integral Quadratic Constraints, Tailored computational robustness tests for complex interconnections

A central notion in systems theory is dissipativity, which has been introduced by Jan Willems with the explicit goal of arriving at a fundamental understanding of the stability properties of feedback interconnections. In robust control, the framework of integral quadratic constraints (IQCs) builds on the seminal contributions of Yakubovich and Zames in the 1960's. It provides a technique for analyzing the stability of an interconnection of some linear system in feedback with a whole class of systems, also refereed to as uncertainty. In this paper we survey the key ideas of exploiting dissipativity and integral quadratic constraints for the computational analysis of robust stability and performance properties of uncertain interconnections in terms of linear matrix inequalities. In particular for dynamic supply rates, the paper revolves around the notion of finite-horizon integral quadratic constraints with a terminal cost. We reveal that this provides a seamless link between the general IQC theorem and dissipativity theory that has been established only rather recently.

math.OC↗

arXiv2021

Convex Synthesis of Accelerated Gradient Algorithms

We present a convex solution for the design of generalized accelerated gradient algorithms for strongly convex objective functions with Lipschitz continuous gradients. We utilize integral quadratic constraints and the Youla parameterization from robust control theory to formulate a solution of the algorithm design problem as a convex semi-definite program. We establish explicit formulas for the optimal convergence rates and extend the proposed synthesis solution to extremum control problems.

math.OC↗

arXiv2019

Robust and structure exploiting optimization algorithms: An integral quadratic constraint approach

We consider the problem of analyzing and designing gradient-based discrete-time optimization algorithms for a class of unconstrained optimization problems having strongly convex objective functions with Lipschitz continuous gradient. By formulating the problem as a robustness analysis problem and making use of a suitable adaptation of the theory of integral quadratic constraints, we establish a framework that allows to analyze convergence rates and robustness properties of existing algorithms and enables the design of novel robust optimization algorithms with prespecified guarantees capable of exploiting additional structure in the objective function.

math.OC↗