MATH5185 is a Special Topic in Applied Mathematics course for Honours and Postgraduate Coursework students. The current topic is Nonnegativity and polynomial optimization. See the course overview below.

**Units of credit:** 6

**Cycle of offering:** Topics rotate; Term 3, 2022.

**Graduate attributes:** The course will enhance your research, inquiry and analytical thinking abilities.

**More information:** The course outline (pdf) contains information about course objectives, assessment, course materials and the syllabus, and will be provided closer to the start of term.

If you are currently enrolled in MATH5185, you can log into UNSW Moodle for this course.

#### Course Overview

Polynomial optimization is a high-impact area for engineering and computational mathematics, which holds the key to some fundamental problems of discrete optimization, power engineering, theoretical computer science, and dynamics and control. A polynomial optimization problem is a special class of nonconvex nonlinear global optimization in which both the objective and constraints are polynomials. That is, it aims at finding the global minimizer(s) of a multivariate polynomial on a certain set. Polynomial optimization has a wide range of applications like dynamical systems, robotics, computer vision, signal processing, and economics. Specific examples include computing real-time certificates of collision avoidance for drone aircraft and autonomous cars navigating through a cluttered environment or the optimal power flow problem, which consists in computing the best operating point of a power network. Mathematically, it is well-known that solving polynomial optimization is very hard in general. Interestingly, foundations of this problem trace back to the 19th century and Hilbert’s 17th problem since it is intimately related to the problem of recognizing nonnegativity of a multivariate polynomial. One of the most powerful approaches for handling polynomial optimization problems with rigorous and global guarantees is via algebraic techniques. The majority of these techniques use the so-called sum of squares relaxation, which relies on semidefinite programs.

Studying these algebraic techniques form an exciting area of optimization that combines classical concepts of algebraic geometry with modern tools of numerical optimization. In this course, we approach this interesting area from both an applied and a theoretical point of view. This course aims to introduce core statements of real algebraic geometry and its relation to polynomial optimization and to provide an overview of the common methods in polynomial optimization in theory and practice. Topics may include:

- Nonnegative polynomials and sums of squares (SOS)
- Semidefinite optimization: reference to SOS, moments, spectrahedra
- Positivstellensaetze: the basics of polynomial optimization under constraints - Polynomial optimization in practice: Software and solvers; applications (drawn from, e.g., operations research, statistics and machine learning, finance, economics, and engineering)

*The course varies when offered. The learning outcomes upon completion are described below.*

*Demonstrate mastery of an advanced topic in Applied Mathematics.**Display advanced competency in mathematical presentation and written skills.**Demonstrate an ability to apply advanced mathematical techniques to formulate and solve real-world problems.*