Introduction to Nonsmooth Analysis and Optimization

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. The required background from functional analysis and calculus of variations is also briefly summarized.Comment: arXiv admin note: substantial text overlap with arXiv:1708.0418

Forward-Half-Reflected-Partial inverse-Backward Splitting Algorithm for Solving Monotone Inclusions

In this article, we proposed a method for numerically solving monotone inclusions in real Hilbert spaces that involve the sum of a maximally monotone operator, a monotone-Lipschitzian operator, a cocoercive operator, and a normal cone to a vector subspace. Our algorithm splits and exploits the intrinsic properties of each operator involved in the inclusion. The proposed method is derived by combining partial inverse techniques and the {\it forward-half-reflected-backward} (FHRB) splitting method proposed by Malitsky and Tam (2020). Our method inherits the advantages of FHRB, equiring only one activation of the Lipschitzian operator, one activation of the cocoercive operator, two projections onto the closed vector subspace, and one calculation of the resolvent of the maximally monotone operator. Furthermore, we develop a method for solving primal-dual inclusions involving a mixture of sums, linear compositions, parallel sums, Lipschitzian operators, cocoercive operators, and normal cones. We apply our method to constrained composite convex optimization problems as a specific example. Finally, in order to compare our proposed method with existing methods in the literature, we provide numerical experiments on constrained total variation least-squares optimization problems. The numerical results are promising