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This research-oriented course will focus on algebraic and computational techniques for optimization problems involving polynomial equations and inequalities with particular emphasis on the connections with semidefinite optimization.
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Description
The course will develop in a parallel fashion several algebraic and numerical approaches to polynomial systems, with a view towards methods that simultaneously incorporate both elements. We will study both the complex and real cases, developing techniques of general applicability, and stressing convexity-based ideas, complexity results, and efficient implementations. Although we will use examples from several engineering areas, particular emphasis will be given to those arising from systems and control applications.
Course content
- Introduction Unlimited
- PSD Matrices Unlimited
- Binary Optimization Unlimited
- Review: Groups, Rings, Fields Polynomials and Ideals Unlimited
- Univariate Polynomials Unlimited
- Resultants Discriminants Unlimited
- Hyperbolic Polynomials Unlimited
- Relating SDP-representable Sets and Hyperbolic Polynomials Unlimited
- Binomial Equations Unlimited
- Nonegativity and Sums of Squares Unlimited
- SOS Applications Unlimited
- Recovering a Measure from its Moments Unlimited
- Polynomial Ideals Unlimited
- Monomial Orderings Unlimited
- Zero-dimensional Ideals Unlimited
- Generalizing the Hermite Matrix Parametric Versions Unlimited
- Infeasibility of Real Polynomial Equations Certificates Unlimited
- Quantifier Elimination Unlimited
- Certificates Unlimited
- Positive Polynomials Unlimited
- Groups and their Representations Unlimited
- Sums of Squares Programs and Polynomial Inequalities Unlimited
Instructor
