This book presents a comprehensive series of methods in nonsmooth optimization, with a particular focus on their application in stochastic programming and dedicated algorithms for decision-making under uncertainty. Each method is accompanied by rigorous mathematical analysis, ensuring a deep understanding of the underlying principles. The theoretical discussions included are essential for comprehending the mechanics of various algorithms and the nature of the solutions they providewhether they are global, local, stationary, or critical. The book begins by introducing fundamental tools from set-valued analysis, optimization, and probability theory. It then transitions from deterministic to stochastic optimization, starting with a thorough discussion of modeling, understanding uncertainty, and incorporating it into optimization problems. Following this foundation, the book explores numerical algorithms for nonsmooth optimization, covering well-known decomposition techniques and algorithms for convex optimization, mixed-integer convex programming, and nonconvex optimization. Additionally, it introduces numerical algorithms specifically for stochastic programming, focusing on stochastic programming with recourse, chance-constrained optimization, and detailed algorithms for both risk-neutral and risk-averse multistage stochastic programs.

The book guides readers through the entire process, from defining optimization models for practical problems to presenting implementable algorithms that can be applied in practice. It is intended for students, practitioners, and scholars who may be unfamiliar with stochastic programming and nonsmooth optimization. The analyses provided are also valuable for practitioners who may not be interested in convergence proofs but wish to understand the nature of the solutions obtained.

Presents advanced methods in nonsmooth optimization and stochastic programming Offers step-by-step guidance with rigorous mathematical analysis Introduces practical, implementable algorithms

Autorentext

Wim van Ackooij holds a PhD degree from École Centrale de Paris and a Habilitation from Université Paris 1 Panthéon-Sorbonne, France, both in Applied Mathematics. He is Associate Editor of Optimization and Mathematical Programming Computation. Wim has published nearly 70 papers in refereed journals and has extensive experience in stochastic optimization, specifically probabilistically constrained programming, as well as unit commitment and bundle methods. He has also worked on practical applications of optimization in the energy industry for over 20 years. Welington de Oliveira is an Associate Professor at the Centre de Mathématiques Appliquées, Mines Paris - PSL, France. He obtained his PhD in systems engineering and computer science from the Federal University of Rio de Janeiro, Brazil, and has a Habilitation in applied mathematics from Université Paris 1 Panthéon Sorbonne, France. Welington has extensive experience in nonsmooth optimization and stochastic programming, having published numerous research articles and served as an associate editor for several reputable journals in the field.

Inhalt

Introduction.- Primer of convex analysis.- Variational analysis.- Linear and nonlinear optimization problems.- Probability and Statistics.- Fundamental modeling questions in stochastic programming.- Adjusting to uncertainty: modeling recourse.- Probability constraints.- Proximal point algorithms for problems with structure.- Cutting-plane algorithms for nonsmooth convex optimization over simple domains.- Bundle methods for nonsmooth convex optimization over simple domains.- Methods for nonlinearly constrained nonsmooth optimization problems.- Methods for nonsmooth optimization with mixed-integer variables.- Methods for nonsmooth nonconvex optimization.- Two-stage stochastic programs.- Progressive decoupling in multistage stochastic programming.- Scenario decomposition with alternating projections.- Methods for multistage stochastic linear programs.- Methods for handling probability.