# A Comparison of Nonsmooth, Nonconvex, Constrained Optimization Solvers for the Design of Time-Delay Compensators

## TL;DR

This paper compares the performance of two advanced solvers for designing time-delay compensators, addressing the challenges of nonsmooth, nonconvex constrained optimization in robust control, and introduces new visualization tools for benchmarking.

## Contribution

It provides a detailed performance comparison of solvers on complex control design problems and introduces novel visualization tools for benchmarking constrained optimization methods.

## Key findings

- Benchmarking results highlight differences in solver performance.
- Introduction of Global-Local Profiles for optimization tradeoff analysis.
- Application to robust control design with nonsmooth, nonconvex constraints.

## Abstract

We present a detailed set of performance comparisons of two state-of-the-art solvers for the application of designing time-delay compensators, an important problem in the field of robust control. Formulating such robust control mechanics as constrained optimization problems often involves objective and constraint functions that are both nonconvex and nonsmooth, both of which present significant challenges to many solvers and their end-users hoping to obtain good solutions to these problems. In our particular engineering task, the main difficulty in the optimization arises in a nonsmooth and nonconvex stability constraint, which states that the infinite spectrum of zeros of the so-called shaper should remain in the open left half-plane. To perform our evaluation, we make use $\beta$-relative minimization profiles, recently introduced visualization tools that are particularly suited for benchmarking solvers on nonsmooth, nonconvex, constrained optimization problems. Furthermore, we also introduce new visualization tools, called Global-Local Profiles, which for a given problem and a fixed computational budget, assess the tradeoffs of distributing the budget over few or many starting points, with the former getting more budget per point and latter less.