Semidefiniete Optimalisering

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Aim of the course

The course is aimed at students with an interest in optimization, combinatorics, geometry, and algebra. The purpose of the course is to give an introduction to the theory, computational techniques, and applications of semidefinite optimization. In particular, after successful participation in the course, students will be able to: explain the theory and algorithmic approach to solve semidefinite optimization problems, give examples of problems in optimization, combinatorics, geometry, and algebra to which semidefinite optimization is applicable, solve semidefinite optimization problems with the help of solvers, and recognize problems that can be tackled using semidefinite optimization.

Semidefinite optimization is a recent tool in mathematical optimization and can be seen as a vast generalization of linear programming. One can define it as minimizing a linear function of a symmetric, positive semidefinite matrix subject to linear constraints. Only a few decades ago it became clear that one can solve semidefinite optimization problems efficiently in theory and practice. Since then, semidefinite optimization has become a frequently used tool of high mathematical elegance and computational power.

Contents:

• Part 1 (Theory of semidefinite optimization): conic programming, duality theory, algorithms (only selected topics)

• Part 2 (Applications in combinatorics): Lovász theta function, 0/1 programming, max-cut, Grothendieck’s constant

• Part 3 (Applications in geometry): geometry of spectrahedra, hidden convexity results, kissing number, sphere packings

• Part 4 (Applications in algebra): polynomial optimization, positive polynomials and sums of squares, Lasserre hierarchy, noncommutative setting and applications to quantum information.