Nonsmooth Analysis and Optimization

Abstract

These lecture notes for a graduate course cover generalized derivative concepts useful in deriving necessary optimality conditions and numerical algorithms for nondifferentiable optimization problems in inverse problems, imaging, and PDE-constrained optimization. Treated are convex functions and subdifferentials, Fenchel duality, monotone operators and resolvents, Moreau--Yosida regularization, proximal point and (some) first-order splitting methods, Clarke subdifferentials, and semismooth Newton methods. The required background from functional analysis and calculus of variations is also briefly summarized.

arXiv commentLecture notes

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• Finite Codimensional Controllability, and Optimal Control Problems with Endpoint State Constraints
• A proximal dual semismooth Newton method for computing zero-norm penalized QR estimator
• An Efficient Augmented Lagrangian Method with Semismooth Newton Solver for Total Generalized Variation
• Convex Analysis of Minimal Time and Signed Minimal Time Functions