Introduction to Nonsmooth Analysis and Optimization

TL;DR

This book provides a comprehensive, discretization-agnostic treatment of nonsmooth optimization in infinite-dimensional spaces by unifying convex and nonconvex analysis with set-valued calculus. It develops the foundational functional-analytic tools (normed and Hilbert spaces, duality, weak convergence) and builds from convex subdifferentials to general set-valued derivatives, augmented by proximal- and Newton-type algorithms. The text also integrates variational principles, Γ-convergence, and Fenchel duality to derive explicit optimality conditions and stable, scalable computational schemes, with applications to inverse problems, imaging, and PDE-constrained optimization. The framework aims to bridge theory and numerics, enabling robust analysis and discretization-independent convergence results for nonsmooth problems in function spaces.

Abstract

This book aims to give an introduction to generalized derivative concepts useful in deriving necessary optimality conditions and numerical algorithms for infinite-dimensional nondifferentiable optimization problems that arise in inverse problems, imaging, and PDE-constrained optimization. They cover convex subdifferentials, Fenchel duality, monotone operators and resolvents, Moreau--Yosida regularization as well as Clarke and (briefly) limiting subdifferentials. Both first-order (proximal point and splitting) methods and second-order (semismooth Newton) methods are treated. In addition, differentiation of set-valued mapping is discussed and used for deriving second-order optimality conditions for as well as Lipschitz stability properties of minimizers. Applications to inverse problems and optimal control of partial differential equations illustrate the derived results and algorithms. The required background from functional analysis and calculus of variations is also briefly summarized.

Figures (43)

• Figure 1: The polar cone $A^{\circ}$ is is bounded by the half-lines at right angles to the smallest cone containing $A$.
• Figure 2: Illustration of lower semicontinuity: two functions $F_1,F_2:\mathbb{R}\to\mathbb{R}$ and a sequence $\{x_n\}_{n\in\mathbb{N}}$ realizing their (identical) limes inferior.
• Figure 3: Illustration of a convex set and of the characterization of a convex function in terms of the convexity of its epigraph: all line segments between two points of corresponding set are completely contained in that set.
• Figure 4: Illustration of $\mathop{\mathrm{\mathrm{graph}}}\nolimits \partial F$ for two different functions $F:\mathbb{R}\to\overline{\mathbb{R}}$.
• Figure 5: Normal cones of a convex set $C$ at two points $x_1$ and $x_2$.

Theorems & Definitions (890)

• Example 1.1
• Lemma 1.2: core--int
• Example 1.3
• Theorem 1.4: Hahn--Banach, analytic
• Theorem 1.5: Hahn--Banach, geometric
• Corollary 1.6
• proof
• Corollary 1.7
• proof
• Theorem 1.8: bipolar