Derivative-free methods for nonsmooth optimization (DFNO)

Many real-life tasks in science and engineering require solving optimization problems whose behavior can change abruptly. These problems are often nonsmooth, meaning that their objective functions or constraints may have sharp bends or sudden jumps. In such situations, the usual mathematical information needed by classical optimization methods, such as derivatives or subgradients, is either extremely difficult or too costly to compute. As a result, traditional approaches struggle with many important applications. This project develops mathematical methods that can handle nonsmooth problems without relying on derivatives. These derivative-free techniques are useful when the objective value comes from a black-box source, such as an expensive measurement, a simulation, or another procedure that does not reveal its internal structure. The goal is to design new algorithms that can work reliably under various types of restrictions, including continuous, integer, and mixed-integer variables, as well as bound, linear, and nonlinear constraints. The project will create several families of methods. Some of them estimate subgradients in a structured way and combine this information with low-dimensional strategies. Others follow accelerated procedures that allow faster progress while still avoiding the need for derivative or subgradient calculations. All methods aim to be efficient, robust, and broadly applicable. To ensure practical relevance, the algorithms will be tested on many benchmark problems and real-world examples, including those from machine learning. Their performance will be compared with that of established solvers. The project is expected to provide both new theoretical insights and user-friendly software that supports researchers and practitioners who work with challenging nonsmooth optimization problems.