Functions that are not differentiable in the classical sense have become a central tool in modern mathematical models for imaging, inverse problems, machine learning, and optimal control of differential equations. These models are increasingly formulated in infinite-dimensional function spaces to be independent of problem size and discretization quality. Introduction to Nonsmooth Analysis and Optimization presents a unified and rigorous introduction to the infinite-dimensional analysis and algorithmic solution of nonsmooth optimization problems arising from the above-mentioned models, including the necessary theoretical tools of nonsmooth analysis to state-of-the-art algorithms and their convergence and stability analysis.Introduction to Nonsmooth Analysis and Optimization offersa thorough examination of analysis and algorithms—first- and second-order methods— in infinite dimensions,a self-contained and accessible introduction to set-valued and variational analysis for optimization problems and includes novel calculus results for relevant situations.Julia code to replicate the numerical results.