Handbook on semidefinite, conic and polynomial optimization

Online-Ressource (XI, 960 S.)

International series in operations research & management science ; 166

Description based upon print version of record

Handbook on Semidefinite, Conic and Polynomial Optimization; Preface; Contents; 1 Introduction to Semidefinite, Conic and Polynomial Optimization; 1.1 Semidefinite Programming; 1.2 Positive Semidefinite Matrices; 1.3 Duality in SDP; 1.4 Computational Complexity and SDP; 1.5 Introduction; 1.5.1 Two Dual Points of View; 1.6 Moments and Positive Polynomials; 1.6.1 Moment Matrix; 1.6.2 Localizing Matrix; 1.6.3 Certificates of Positivity; 1.7 A Hierarchy of Semidefinite Relaxations; 1.7.1 Nonlinear 0/1 Programs; 1.7.2 Handling Large Size Problems; 1.7.3 Software; References; Part I Theory

2 The Approach of Moments for Polynomial Equations2.1 Introduction; 2.1.1 Existing Methods; 2.1.1.1 Existing Methods over the Complex Numbers; 2.1.1.2 Existing Methods over the Real Numbers; 2.1.2 The Basic Idea of the Moment Method; 2.1.3 Organization of the Chapter; 2.2 Preliminaries of Polynomial Algebra; 2.2.1 Polynomial Ideals and Varieties; 2.2.1.1 The Polynomial Ring and Its Dual; 2.2.1.2 Ideals and Varieties; 2.2.2 The Eigenvalue Method for Complex Roots; 2.2.2.1 The Quotient Space bold0mu mumu R[x]/IR[x]/IR[x]/IR[x]/IR[x]/IR[x]/I; 2.2.2.2 Computing Roots with Companion Matrices

2.2.3 Border Bases and Normal Forms2.3 The Moment Method for Real Root Finding; 2.3.1 Positive Linear Forms and Real Radical Ideals; 2.3.2 Truncated Positive Linear Forms and Real Radical Ideals; 2.3.3 The Moment Matrix Algorithm for Computing Real Roots; 2.3.4 Real vs. Complex Root Finding; 2.4 Further Directions and Connections; 2.4.1 Optimization and Polynomial Inequalities; 2.4.2 Exact Certificates of Emptiness; 2.4.3 Positive Dimensional Ideals and Quotient Ideals; References; 3 Algebraic Degree in Semidefinite and Polynomial Optimization; 3.1 Introduction; 3.2 The Optimization Problem

3.3 The Derivation of Algebraic Degree3.3.1 The Complex Projective Setup; 3.3.2 Genericity Assumptions and KKT; 3.3.2.1 Necessary Conditions; 3.3.3 Algebra Geometric Methods; 3.3.4 Degree Formulas; 3.3.4.1 POP: The General Polynomial Optimization Problem; 3.3.4.2 Example; 3.3.4.3 QCQP: Quadratically Constrained Quadratic Programming; 3.3.4.4 SDP: Semidefinite Programming; 3.3.4.5 Example; References; 4 Semidefinite Representation of Convex Sets and Convex Hulls; 4.1 Introduction; 4.1.1 LMI Representations; 4.1.2 SDP Representations; 4.1.3 Convex Hulls of Semialgebraic Sets

4.1.4 Constructions and Theory4.1.5 Motivation; 4.1.6 Notations; 4.2 LMI Representation; 4.2.1 Necessary Conditions; 4.2.1.1 Real Zero Condition; 4.2.1.2 Rigid Convexity; 4.2.2 The Main Result About LMI Representability; 4.2.3 Determinantal Representations; 4.3 SDP Representation; 4.3.1 The Main Result for SDP Representability; 4.3.2 Convex Hulls of Semialgebraic Sets; 4.3.3 Necessary Conditions for SDP Representability; 4.4 Moment Type Constructions; 4.4.1 The Conceptual Idea; 4.4.2 A Basic Moment Relaxation; 4.4.3 Refined Moment Type Relaxations; 4.5 Sos-Convex Sets

4.6 Strictly Convex Sets

10.1007/978-1-4614-0769-0 [DOI]

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