## Overview

MATH5185 is a special topic in applied mathematics course for honours and postgraduate coursework students. The Topic title is Nonnegativity and Polynomial Optimization

Units of credit: 6

Cycle of offering: Topics rotate; Term 3

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

The course outline (pdf) contains information about course objectives, assessment, course materials and the syllabus.

Important additional information as of 2023

#### UNSW Plagiarism Policy

The University requires all students to be aware of its policy on plagiarism.

For courses convened by the School of Mathematics and Statistics no assistance using generative AI software is allowed unless specifically referred to in the individual assessment tasks.

If its use is detected in the no assistance case, it will be regarded as serious academic misconduct and subject to the standard penalties, which may include 00FL, suspension and exclusion.

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

## Course overview

Polynomial optimisation is a high-impact area for engineering and computational mathematics, which holds the key to some fundamental problems of discrete optimisation, power engineering, theoretical computer science, and dynamics and control. A polynomial optimisation problem is a special class of nonconvex nonlinear global optimisation in which both the objective and constraints are polynomials. That is, it aims at finding the global minimiser (s) of a multivariate polynomial on a certain set. Polynomial optimisation 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 optimisation 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 recognising nonnegativity of a multivariate polynomial. One of the most powerful approaches for handling polynomial optimisation 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 optimisation that combines classical concepts of algebraic geometry with modern tools of numerical optimisation. 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 optimisation and to provide an overview of the common methods in polynomial optimisation in theory and practice. Topics may include:

• Nonnegative polynomials and sums of squares (SOS)
• Semidefinite optimisation: reference to SOS, moments, spectrahedra
• Positivstellensaetze: the basics of polynomial optimisation under constraints - Polynomial optimisation 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.