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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.


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.


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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
Submit Your Assignment 00:00:00
Certification 00:00:00

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