Advancing Parametric Optimization: On Multiparametric Linear Complementarity Problems with Parameters in General Locations (SpringerBriefs in Optimization)

✍ Scribed by Nathan Adelgren

• Publisher: Springer
• Year: 2021
• Tongue: English
• Leaves: 118
• Edition: 1st ed. 2021
• Category: Library

✦ Synopsis

The theory presented in this work merges many concepts from mathematical optimization and real algebraic geometry. When unknown or uncertain data in an optimization problem is replaced with parameters, one obtains a multi-parametric optimization problem whose optimal solution comes in the form of a function of the parameters.The theory and methodology presented in this work allows one to solve both Linear Programs and convex Quadratic Programs containing parameters in any location within the problem data as well as multi-objective optimization problems with any number of convex quadratic or linear objectives and linear constraints. Applications of these classes of problems are extremely widespread, ranging from business and economics to chemical and environmental engineering. Prior to this work, no solution procedure existed for these general classes of problems except for the recently proposed algorithms

✦ Table of Contents

Preface
Acknowledgments
Contents
Nomenclature
1 Introduction
2 Background on mpLCP
2.1 Preliminaries
2.2 Invariancy Regions
2.3 Geometry of the mpLCP
3 Algebraic Properties of Invariancy Regions
3.1 Decomposition of the Parameter Space
3.2 Exploiting the Algebraic Structure of an Invariancy Region
3.3 An Initial Strategy for Partitioning the Parameter Space
4 Phase 2: Partitioning the Parameter Space
4.1 Computing Adjacent Bases
4.2 Computing Adjacent Invariancy Regions
4.2.1 Construction of Special Sets
4.2.2 Revisiting Examples
4.2.3 Determining Adjacent Invariancy Regions Given a Full Dimensional Region
4.2.4 Identification of Full Dimensional Invariancy Regions
4.2.5 Determining Adjacent Invariancy Regions from a Given (k-1)-Dimensional Region
5 Phase 1: Determining an Initial Feasible Solution
6 Further Considerations
6.1 On the Importance of Assumptions 1.1 and 1.2
6.2 On Obtaining Non-overlapping Invariancy Regions
7 Assessment of Performance
7.1 Experimental Results
7.2 Computational Complexity
8 Conclusion
A Tableaux for Example 2.1
B Tableaux for Example 2.2
References