An Introduction to Nonsmooth Analysis

• 1. Basic concepts and results: Upper and lower limits. Semicontinuity. Differentiability. Two important Theorems.2. Convex Functions: Convex sets and convex functions. Continuity of convex functions. Separation Results. Convexity and Differentiability. 3. The subdifferential of a Convex function: Subdifferential properties. Examples.4. The subdifferential. General case: Definition and basic properties. Geometrical meaning of the subdifferential. Density of subdifferentiability points. Proximal subdifferential 5. Calculus: Sum Rule. Constrained minima. Chain Rule. Regular functions: Elementary properties. Mean Value results. Decreasing Functions 6. Lipschitz functions and the generalized gradient: Lipschitz regular functions. The generalized gradient. Generalized Jacobian. Graphical derivative 7. Applications: Flow invariant sets. Viscosity solutions. Solving equations.

"...starting from the very beginning, adopting a slow, easy to follow linear development and reaching to a self-contained theory...oriented towards undergraduate students, as a first quick introduction to the topic." --MathSciNet"...devoted to presenting the theory of the subdifferential of lower semicontinuous functions which is a generalization of the subdifferential of convex functions...a good reference for researchers in optimization and applied mathematics." --Zentralblatt MATH, Sep-14