2. Introduction to Semidefinite Programs Masakazu Kojima Semidefinite Programming and Its Applications Institute for Mathematical Sciences National University of Singapore Jan 9 -13, 2006

3. Main purpose Introduction of semidefinite programs Brief review of SDPs 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 Contents References --- Not comprehensive but helpful for further study of the subject ---

4. 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

5. 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

6. 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

13. Classification of Optimization Problems ContinuousDiscreteConvex Nonconvex Linear Optimization Problem over Symmetric Cone SemiDefinite Program Second Order Cone Program Convex Quadratic Optimization Problem Linear Program Polynomial Optimization Problem 0-1 Integer LP & QOP relaxation POP over Symmetric Cone Bilinear Matrix Inequality | | | |

14. 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

37. 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 product 5. General SDPs 6. Some examples 7. Duality

44. 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

83. 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

85. 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

90. 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

111. 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

134. References