Introduction to Semidefinite Programs

Masakazu Kojima

Semidefinite Programming and Its Applications

Institute for Mathematical Sciences

National University of SingaporeJan 9 -13, 2006

Main purpose

• Introduction of semidefinite programs

• Brief review of SDPs

Part I: Introduction to SDP and its basic theory --- 70 minutesPart II: Primal-dual interior-point methods --- 70 minutesPart III: Some applications Appendix: Linear optimization problems over symmetric cones

Contents

References--- Not comprehensive but helpful for further study of the subject ---

ContentsPart I: Introduction to SDP and its basic theory1. LP versus SDP 2. 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 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

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

Part I: Introduction to SDP and its basic theory

1. LP versus SDP 2. 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 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 product5. General SDPs6. Some examples7. Duality

Part I: Introduction to SDP and its basic theory

1. LP versus SDP 2. 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

Continuous DiscreteConvex

NonconvexLinear Optimization Problem over Symmetric Cone

SemiDefinite Program

Second Order Cone Program

Convex Quadratic Optimization Problem

Linear Program

Polynomial OptimizationProblem

] [←

]

]

−−−−→

0-1 IntegerLP & QOP

relaxation U

U

U

U

U

POP overSymmetric Cone

Bilinear　MatrixInequality

||||

I

U

U

I−−→

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 product5. General SDPs6. Some examples7. Duality

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 product5. General SDPs6. Some examples7. Duality

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 SDPs6. Some examples7. Duality

Part I: Introduction to SDP and its basic theory

1. LP versus SDP 2. Why is SDP interesting and important? 3. The equality standard form4. Some basic properties on positive semidefinite matrices and their inner product5. General SDPs

6. Some examples7. Duality

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 SDP 4. Some basic properties on positive semidefinite matrices and their inner product5. General SDPs6. Some examples

7. Duality

Part II: Primal-dual interior-point methods

1. 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

Part II: Primal-dual interior-point methods

1. 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

Part II: Primal-dual interior-point methods

1. Existing numerical methods for SDPs

2. 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

Part II: Primal-dual interior-point methods

1. 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

Part II: Primal-dual interior-point methods

1. Existing numerical methods for SDPs2. Three approaches to primal-dual interior-point methods for SDPs3. The central trajectory

4. Search directions5. Various primal-dual interior-point methods6. Exploiting sparsity7. Software packages8. SDPA sparse format9. Numerical results

Part II: Primal-dual interior-point methods

1. Existing numerical methods for SDPs2. Three approaches to primal-dual interior-point methods for SDPs3. The central trajectory4. Search directions

5. Various primal-dual interior-point methods6. Exploiting sparsity7. Software packages8. SDPA sparse format9. Numerical results

Part II: Primal-dual interior-point methods

1. 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

Part II: Primal-dual interior-point methods

1. 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 sparsity

7. Software packages8. SDPA sparse format9. Numerical results

Part II: Primal-dual interior-point methods

1. 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 packages

8. SDPA sparse format9. Numerical results

Part II: Primal-dual interior-point methods

1. 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 format

9. Numerical results

Part III: Some applications

1. Matrix approximation problems2. A nonconvex quadratic optimization problem3. The max-cut problem4. Sum of squares of polynomials

Part III: Some applications

1. Matrix approximation problems2. A nonconvex quadratic optimization problem3. The max-cut problem4. Sum of squares of polynomials

Part III: Some applications

1. Matrix approximation problems

2. A nonconvex quadratic optimization problem3. The max-cut problem4. Sum of squares of polynomials

Part III: Some applications

1. Matrix approximation problems2. A nonconvex quadratic optimization problem

3. The max-cut problem4. Sum of squares of polynomials

N ={1,2,3, 4,5,6, 7} , w12 =w21 =2,...

•

•

•

•

•

•

••

K ={1,2,7} ⇒ δ(K ) ={{2,3},{3,7} ,{6,7}} w(δ(K )) =7 + 4 + 5 =16K ={1,2,3,4,6} ⇒ δ(K ) ={{1,7},{2,7} ,{3,7} ,{4,5} ,{5,6} ,{6,7}} w(δ(K )) =3+ 5 + 4 + 7 + 8 + 5 =32

N ={1,2,3, 4,5,6, 7} , w12 =w21 =2,...

•

•

•

•

•

•

•

•

K ={1,2,7} ⇒ δ(K ) ={{2,3},{3,7},{6,7}} w(δ(K )) =7 + 4 + 5 =16K ={1,2,3,4,6} ⇒ δ(K ) ={{1,7},{2,7},{3,7},{4,5},{5,6},{6,7}} w(δ(K )) =3+ 5 + 4 + 7 + 8 + 5 =32

Part III: Some applications

1. Matrix approximation problems2. A nonconvex quadratic optimization problem3. The max-cut problem

4. Sum of squares of polynomials

Appendix: Linear optimization problems over symmetric cones

1. 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 cones

1. 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 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

Appendix: Linear optimization problems over symmetric cones

1. 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 cones

1. Linear optimization problems over cones2. Symmetric cones 3. Euclidean Jordan algebra 4. SOCP (Second Order Cone Program)5. Some applications of SOCPs

If X,Y * OthenX : Y= 0 , XY= O;

X Y= 21(XY + YX) = Oo

Appendix: Linear optimization problems over symmetric cones

1. Linear optimization problems over cones2. Symmetric cones 3. Euclidean Jordan algebra 4. SOCP (Second Order Cone Program)

5. Some applications of SOCPs

References