Topics in Convex Optimisation (Michaelmas 2018)

References

• A. Ben-Tal and A. Nemirovski, Lectures on Modern Convex Optimization (SIAM).
• G. Blekherman, P. A. Parrilo, R. R. Thomas (editors), Semidefinite Optimization and Convex Algebraic Geometry (SIAM).
• S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press.

Lecture notes

• Lecture 1: Review of convexity (separating hyperplane theorem, faces and extreme points, Minkowski theorem)
• Lecture 2: Review of convexity (continued) (convex cones, dual cones)
• Lecture 3: The positive semidefinite cone - Conic programming
• Lecture 4: Semidefinite programming (definition and first examples)
• Lecture 5: Duality in conic programming (1)
• Lecture 6: Duality in conic programming (2) (Strong duality theorem)
• Lecture 7: Binary quadratic optimisation (1) (Max cut)
• Lecture 8: Binary quadratic optimisation (2) (Max cut - continued)
• Lecture 9: The stable set problem and Lovász theta function
• Lecture 10: Nonnegative polynomials on the real line
• Lecture 11: Nonnegative polynomials on the real line (continued)
• Lecture 12: Nonnegative polynomials on the real line (continued)
• Lecture 13: Nonnegative multivariate polynomials
• Lecture 14: Sum of squares hierarchies and Positivstellensatz
• Lecture 15: Sum of squares on the hypercube
• Lecture 16: Sum of squares on the hypercube (continued)

Example classes

• Example class 1: Tuesday 23/10 at 2pm in MR5. Exercise sheet. Solutions.
• Example class 2: Thursday 22/11 at 3pm in MR5. Exercise sheet. Solutions.
• Example class 2: Monday 21/01/2019 at 2pm in MR5. Exercise sheet. Solutions.

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