Rebennack S. Large-scale Optimization 2025

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Rebennack S. Large-scale Optimization 2025

Textbook in PDF format This book provides a comprehensive overview of large-scale and separable optimization. It begins by framing general problems, then focuses on separable structures with complicating constraints or variables, highlighting applications for decomposition techniques like Lagrangian relaxation. A significant portion is dedicated to the fundamentals of Linear Programming (LP), covering problem formulation, convexity, and polyhedral sets. It delivers a detailed examination of the Simplex Method, from its basic mechanics and tableaus to advanced topics like initialization, degeneracy, cycling prevention using Lexicographic and Bland's rules, and its computational complexity. The section concludes by introducing the critical concepts of duality and sensitivity analysis. Separable Optimization Problems and Motivating Examples General Optimization Problem Separable Optimization Problem Decentralized Structures Complicating Constraints Complicating Variables More Fields of Application for Decomposition Techniques Lagrangian Relaxation for Linear Programs The Simplex Method (matrix version) Fundamentals The Linear Programming Problem Standard and Canonical Form Convex Sets and Convex Functions Polyhedral Sets and Their Representation Linear Programming and Convexity The Simplex Method Towards the Simplex Method The Basic Simplex Method Finite Convergence (in the Absence of Degeneracy) Block Pivoting Tableau Starting Solution and Convergence The Initial Basic Feasible Solution Easy Case Artificial Variables Degeneracy, Cycling, and Stalling Lexicographic Rule for Selecting an Exiting Variable Bland’s Rule for Selecting Entering and Exiting Variables Stalling Complexity of the Simplex Algorithm (Klee-Minty) Duality and Sensitivity Analysis Dual Problem Primal-Dual Relationships The Dual Simplex Method The Primal-Dual Method Column Generation Main Idea Master problem and restricted master problem Pricing Step Dual Problem of (RMP) Pricing Problem Algorithm Correctness Column Generation vs Simplex Method Extended Pricing Problem Algorithm with Extended Pricing Problem Dual Bounds Starting Solution Big-M Method Farkas Pricing Stabilization of column generation Branch-and-Price method Dantzig-Wolfe Decomposition Formal Structure Lectures on Polyhedra Decomposition Technique Restricted Master Problem Pricing Problem Starting Solution Detecting Infeasibility Algorithm Correctness Dual Bounds for Dantzig-Wolfe Decompositon Stabilization of Column Generation via Primal and Dual Strategies Benders Decomposition Introduction The Benders decomposition method References