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This course will focus on algebraic and computational techniques for optimization problems involving polynomial equations and inequalities with emphasis on the connections with semidefinite optimization. It will develop in a parallel fashion several algebraic and numerical approaches to polynomial systems that simultaneously incorporate both elements. It also covers both the complex and real cases, developing techniques of general applicability, and stressing convexity-based ideas.
Assessment
This course does not involve any written exams. Students need to answer 5 assignment questions to complete the course, the answers will be in the form of written work in pdf or word. Students can write the answers in their own time. Each answer needs to be 200 words (1 Page). Once the answers are submitted, the tutor will check and assess the work.
Certification
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Course Credit: MIT
Course Curriculum
| Module 01 | |||
| Introduction | 00:10:00 | ||
| PSD Matrices | 00:10:00 | ||
| Binary Optimization | 00:10:00 | ||
| Groups, Rings, Fields | 00:10:00 | ||
| Univariate Polynomials | 00:10:00 | ||
| Resultants | 00:10:00 | ||
| Hyperbolic Polynomials | 00:10:00 | ||
| SDP Representability | 00:10:00 | ||
| Binomial Equations | 00:10:00 | ||
| Nonegativity and Sums of Squares | 00:10:00 | ||
| SOS Applications | 00:10:00 | ||
| Module 02 | |||
| Recovering a Measure from its Moments | 00:10:00 | ||
| Polynomial Ideals | 00:10:00 | ||
| Monomial Orderings | 00:10:00 | ||
| Zero-dimensional Ideals | 00:10:00 | ||
| Generalizing the Hermite Matrix | 00:10:00 | ||
| Infeasibility of Real Polynomial Equations | 00:10:00 | ||
| Quantifier Elimination | 00:10:00 | ||
| Certificates | 00:10:00 | ||
| Positive Polynomials | 00:10:00 | ||
| Groups and their Representations | 00:10:00 | ||
| Sums of Squares Programs and Polynomial Inequalities | 00:10:00 | ||
| Assessment | |||
| Submit Your Assignment | 00:00:00 | ||
| Certification | 00:00:00 | ||
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