Introduction to Semidefinite Programs Masakazu Kojima. Semidefinite Programming and Its Applications Institute for Mathematical Sciences National University of Singapore Jan 9 13, 2006. Main purpose. Introduction of semidefinite programs Brief review of SDPs. Contents.
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Introduction to Semidefinite ProgramsMasakazu Kojima
Semidefinite Programming and Its Applications
Institute for Mathematical Sciences
National University of Singapore
Jan 9 13, 2006
Contents
Part I: Introduction to SDP and its basic theory  70 minutes
Part II: Primaldual interiorpoint methods  70 minutes
Part III: Some applications
Appendix: Linear optimization problems over symmetric cones
References
 Not comprehensive but helpful for further study of the subject 
Part I: Introduction to SDP and its basic theory
1. LP versus SDP
2. Why is SDP interesting and important?
3. The equality standard form
4. Some basic properties on positive semidefinite matrices and their inner product
5. General SDPs
6. Some examples
7. Duality
Part II: Primaldual interiorpoint methods
1. Existing numerical methods for SDPs
2. Three approaches to primaldual interiorpoint methods for SDPs
3. The central trajectory
4. Search directions
5. Various primaldual interiorpoint methods
6. Exploiting sparsity
7. Software packages
8. SDPA sparse format
9. Numerical results
Part III: Some applications
1. Matrix approximation problems
2. A nonconvex quadratic optimization problem
3. The maxcut problem
4. Sum of squares of polynomials
Appendix: Linear optimization problems over symmetric cones
1. Linear optimization problems over cones
2. Symmetric cones
3. Euclidean Jordan algebra
4. SOCP (Second Order Cone Program)
5. Some applications of SOCPs
Part I: Introduction to SDP and its basic theory1. LP versus SDP2. Why is SDP interesting and important? 3. The equality standard form4. Some basic properties on positive semidefinite matrices and their inner product5. General SDPs6. Some examples7. Duality
Part I: Introduction to SDP and its basic theory1. LP versus SDP2. Why is SDP interesting and important? 3. The equality standard form 4. Some basic properties on positive semidefinite matrices and their inner product5. General SDPs6. Some examples7. Duality
Part I: Introduction to SDP and its basic theory1. LP versus SDP2. Why is SDP interesting and important? 3. The equality standard form4. Some basic properties on positive semidefinite matrices and their inner product5. General SDPs6. Some examples7. Duality
Classification of Optimization Problems
Convex
Continuous
Discrete
Nonconvex
01 Integer
LP & QOP
Linear Optimization Problem
over Symmetric Cone
relaxation
Polynomial
Optimization
Problem
SemiDefinite Program




Second Order Cone Program
POP over
Symmetric Cone
Convex Quadratic
Optimization Problem
BilinearMatrix
Inequality
Linear Program
Part I: Introduction to SDP and its basic theory1. LP versus SDP2. Why is SDP interesting and important?3. The equality standard form4. Some basic properties on positive semidefinite matrices and their inner product5. General SDPs6. Some examples7. Duality
Part I: Introduction to SDP and its basic theory1. LP versus SDP2. Why is SDP interesting and important? 3. The equality standard form4. Some basic properties on positive semidefinite matrices and their inner product5. General SDPs6. Some examples7. Duality
Part I: Introduction to SDP and its basic theory1. LP versus SDP2. Why is SDP interesting and important? 3. The equality standard form 4. Some basic properties on positive semidefinite matrices and their inner product5. General SDPs6. Some examples7. Duality
Part I: Introduction to SDP and its basic theory1. LP versus SDP2. Why is SDP interesting and important? 3. The equality standard form4. Some basic properties on positive semidefinite matrices and their inner product5. General SDPs6. Some examples7. Duality
Part I: Introduction to SDP and its basic theory1. LP versus SDP2. Why is SDP interesting and important? 3. The equality standard form SDP 4. Some basic properties on positive semidefinite matrices and their inner product5. General SDPs6. Some examples7. Duality
Part II: Primaldual interiorpoint methods1. Existing numerical methods for SDPs2. Three approaches to primaldual interiorpoint methods for SDPs 3. The central trajectory4. Search directions5. Various primaldual interiorpoint methods6. Exploiting sparsity7. Software packages8. SDPA sparse format9. Numerical results
Part II: Primaldual interiorpoint methods1. Existing numerical methods for SDPs2. Three approaches to primaldual interiorpoint methods for SDPs3. The central trajectory4. Search directions5. Various primaldual interiorpoint methods6. Exploiting sparsity7. Software packages8. SDPA sparse format9. Numerical results
Part II: Primaldual interiorpoint methods1. Existing numerical methods for SDPs2. Three approaches to primaldual interiorpoint methods for SDPs3. The central trajectory4. Search directions5. Various primaldual interiorpoint methods6. Exploiting sparsity7. Software packages8. SDPA sparse format9. Numerical results
Part II: Primaldual interiorpoint methods1. Existing numerical methods for SDPs2. Three approaches to primaldual interiorpoint methods for SDPs3. The central trajectory4. Search directions5. Various primaldual interiorpoint methods6. Exploiting sparsity7. Software packages8. SDPA sparse format9. Numerical results
Part II: Primaldual interiorpoint methods1. Existing numerical methods for SDPs2. Three approaches to primaldual interiorpoint methods for SDPs3. The central trajectory4. Search directions5. Various primaldual interiorpoint methods6. Exploiting sparsity7. Software packages8. SDPA sparse format9. Numerical results
Part II: Primaldual interiorpoint methods1. Existing numerical methods for SDPs2. Three approaches to primaldual interiorpoint methods for SDPs3. The central trajectory4. Search directions5. Various primaldual interiorpoint methods6. Exploiting sparsity7. Software packages8. SDPA sparse format9. Numerical results
Part II: Primaldual interiorpoint methods1. Existing numerical methods for SDPs2. Three approaches to primaldual interiorpoint methods for SDPs3. The central trajectory4. Search directions5. Various primaldual interiorpoint methods6. Exploiting sparsity7. Software packages8. SDPA sparse format9. Numerical results
Part II: Primaldual interiorpoint methods1. Existing numerical methods for SDPs2. Three approaches to primaldual interiorpoint methods for SDPs3. The central trajectory4. Search directions5. Various primaldual interiorpoint methods6. Exploiting sparsity7. Software packages8. SDPA sparse format9. Numerical results
Part II: Primaldual interiorpoint methods1. Existing numerical methods for SDPs2. Three approaches to primaldual interiorpoint methods for SDPs3. The central trajectory4. Search directions5. Various primaldual interiorpoint methods.6. Exploiting sparsity7. Software packages8. SDPA sparse format9. Numerical results
Part II: Primaldual interiorpoint methods1. Existing numerical methods for SDPs2. Three approaches to primaldual interiorpoint methods for SDPs3. The central trajectory4. Search directions5. Various primaldual interiorpoint methods6. Exploiting sparsity7. Software packages8. SDPA sparse format9. Numerical results
Part III: Some applications1. Matrix approximation problems2. A nonconvex quadratic optimization problem3. The maxcut problem4. Sum of squares of polynomials
Part III: Some applications1. Matrix approximation problems2. A nonconvex quadratic optimization problem3. The maxcut problem4. Sum of squares of polynomials
Part III: Some applications1. Matrix approximation problems2. A nonconvex quadratic optimization problem3. The maxcut problem4. Sum of squares of polynomials
Part III: Some applications1. Matrix approximation problems2. A nonconvex quadratic optimization problem3. The maxcut problem4. Sum of squares of polynomials
Part III: Some applications1. Matrix approximation problems2. A nonconvex quadratic optimization problem3. The maxcut problem4. Sum of squares of polynomials
Appendix: Linear optimization problems over symmetric cones1. Linear optimization problems over cones2. Symmetric cones 3. Euclidean Jordan algebra 4. SOCP (Second Order Cone Program)5. Some applications of SOCPs
Appendix: Linear optimization problems over symmetric cones1. Linear optimization problems over cones2. Symmetric cones 3. Euclidean Jordan algebra 4. SOCP (Second Order Cone Program)5. Some applications of SOCPs
Appendix: Linear optimization problems over symmetric cones1. Linear optimization problems over cones2. Symmetric cones 3. Euclidean Jordan algebra 4. SOCP (Second Order Cone Program)5. Some applications of SOCPs
Appendix: Linear optimization problems over symmetric cones1. Linear optimization problems over cones2. Symmetric cones3. Euclidean Jordan algebra 4. SOCP (Second Order Cone Program)5. Some applications of SOCPs
Appendix: Linear optimization problems over symmetric cones1. Linear optimization problems over cones2. Symmetric cones 3. Euclidean Jordan algebra4. SOCP (Second Order Cone Program)5. Some applications of SOCPs
Appendix: Linear optimization problems over symmetric cones1. Linear optimization problems over cones2. Symmetric cones 3. Euclidean Jordan algebra 4. SOCP (Second Order Cone Program)5. Some applications of SOCPs