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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 programs masakazu kojima l.jpg

Introduction to Semidefinite ProgramsMasakazu Kojima

Semidefinite Programming and Its Applications

Institute for Mathematical Sciences

National University of Singapore

Jan 9 -13, 2006


Main purpose l.jpg
Main purpose

  • Introduction of semidefinite programs

  • Brief review of SDPs

Contents

Part I: Introduction to SDP and its basic theory --- 70 minutes

Part II: Primal-dual interior-point 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 ---


Contents l.jpg
Contents

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: Primal-dual interior-point methods

1. Existing numerical methods for SDPs

2. Three approaches to primal-dual interior-point methods for SDPs

3. The central trajectory

4. Search directions

5. Various primal-dual interior-point methods

6. Exploiting sparsity

7. Software packages

8. SDPA sparse format

9. Numerical results


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Part III: Some applications

1. Matrix approximation problems

2. A nonconvex quadratic optimization problem

3. The max-cut 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


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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


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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


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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


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Classification of Optimization Problems

Convex

Continuous

Discrete

Nonconvex

0-1 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


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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


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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


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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


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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


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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


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Part II: Primal-dual interior-point methods1. Existing numerical methods for SDPs2. Three approaches to primal-dual interior-point methods for SDPs 3. The central trajectory4. Search directions5. Various primal-dual interior-point methods6. Exploiting sparsity7. Software packages8. SDPA sparse format9. Numerical results


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Part II: Primal-dual interior-point methods1. Existing numerical methods for SDPs2. Three approaches to primal-dual interior-point methods for SDPs3. The central trajectory4. Search directions5. Various primal-dual interior-point methods6. Exploiting sparsity7. Software packages8. SDPA sparse format9. Numerical results


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Part II: Primal-dual interior-point methods1. Existing numerical methods for SDPs2. Three approaches to primal-dual interior-point methods for SDPs3. The central trajectory4. Search directions5. Various primal-dual interior-point methods6. Exploiting sparsity7. Software packages8. SDPA sparse format9. Numerical results


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Part II: Primal-dual interior-point methods1. Existing numerical methods for SDPs2. Three approaches to primal-dual interior-point methods for SDPs3. The central trajectory4. Search directions5. Various primal-dual interior-point methods6. Exploiting sparsity7. Software packages8. SDPA sparse format9. Numerical results


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Part II: Primal-dual interior-point methods1. Existing numerical methods for SDPs2. Three approaches to primal-dual interior-point methods for SDPs3. The central trajectory4. Search directions5. Various primal-dual interior-point methods6. Exploiting sparsity7. Software packages8. SDPA sparse format9. Numerical results


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Part II: Primal-dual interior-point methods1. Existing numerical methods for SDPs2. Three approaches to primal-dual interior-point methods for SDPs3. The central trajectory4. Search directions5. Various primal-dual interior-point methods6. Exploiting sparsity7. Software packages8. SDPA sparse format9. Numerical results


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Part II: Primal-dual interior-point methods1. Existing numerical methods for SDPs2. Three approaches to primal-dual interior-point methods for SDPs3. The central trajectory4. Search directions5. Various primal-dual interior-point methods6. Exploiting sparsity7. Software packages8. SDPA sparse format9. Numerical results


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Part II: Primal-dual interior-point methods1. Existing numerical methods for SDPs2. Three approaches to primal-dual interior-point methods for SDPs3. The central trajectory4. Search directions5. Various primal-dual interior-point methods6. Exploiting sparsity7. Software packages8. SDPA sparse format9. Numerical results


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Part II: Primal-dual interior-point methods1. Existing numerical methods for SDPs2. Three approaches to primal-dual interior-point methods for SDPs3. The central trajectory4. Search directions5. Various primal-dual interior-point methods.6. Exploiting sparsity7. Software packages8. SDPA sparse format9. Numerical results


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Part II: Primal-dual interior-point methods1. Existing numerical methods for SDPs2. Three approaches to primal-dual interior-point methods for SDPs3. The central trajectory4. Search directions5. Various primal-dual interior-point methods6. Exploiting sparsity7. Software packages8. SDPA sparse format9. Numerical results


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Part III: Some applications1. Matrix approximation problems2. A nonconvex quadratic optimization problem3. The max-cut problem4. Sum of squares of polynomials


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Part III: Some applications1. Matrix approximation problems2. A nonconvex quadratic optimization problem3. The max-cut problem4. Sum of squares of polynomials


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Part III: Some applications1. Matrix approximation problems2. A nonconvex quadratic optimization problem3. The max-cut problem4. Sum of squares of polynomials


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Part III: Some applications1. Matrix approximation problems2. A nonconvex quadratic optimization problem3. The max-cut problem4. Sum of squares of polynomials


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Part III: Some applications1. Matrix approximation problems2. A nonconvex quadratic optimization problem3. The max-cut problem4. Sum of squares of polynomials


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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


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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


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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


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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


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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


Slide127 l.jpg

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



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