# ACM41030 Optimization Algorithms

This module introduces students to the theory of optimization, a key tool in modern Applied Mathematics, Operations Research, and Machine Learning. Students will study in depth the key concepts in continuous optimization - both unconstrained, constrained, and global.

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Curricular information is subject to change

Learning Outcomes:

On completion of this module, students should be able to:

1. Formulate standard optimization techniques in continuous optimization, understand the convergence criteria, and implement these methods from scratch;
2. Implement the same methods using standard software packages, understand when these methods will work well and when they won’t;
3. Understand the first-order necessary conditions for optimality in constrained optimization, be able to solve simple problems by hand;
4. Prove the Karush-Kuhn-Tucker conditions;
5. Formulate the Dual Problem in constrained optimization;
5. Understand the need for global optimization, implement a simulated-annealing algorithm.

Indicative Module Content:

Topics covered: Steepest-Descent and Newton-type methods, including analysis of convergence, Trust-region methods, including the construction of solutions of the constrained sub-problem; Numerical implementations of standard optimization methods. Constrained Optimization with equality and inequality constraints, examples motivating the introduction of the Lagrange Multiplier Technique. Necessary first-order optimality conditions, including a derivation of the Karush-Kuhn-Tucker conditions. Farkas’s Lemma and the Separating Hyperplane Theorem. Formulation of the Dual Problem in Constrained Optimization. Introduction to Global Optimization, to include a discussion on Simulated Annealing.

Student Effort Hours:
Student Effort Type Hours
Lectures

36

Specified Learning Activities

24

Autonomous Student Learning

40

Total

100

Approaches to Teaching and Learning:
Lectures, tutorials, problem class, coding sessions. Opportunities for students to assess their own progress through study of model answers to exercises, as well as problem-solving and coding.
Requirements, Exclusions and Recommendations

Not applicable to this module.

Module Requisites and Incompatibles
Not applicable to this module.

Assessment Strategy
Description Timing Open Book Exam Component Scale Must Pass Component % of Final Grade
Exam (In-person): One-hour final exam n/a Standard conversion grade scale 40% No

50

Exam (In-person): Class test - held after midterm break n/a Standard conversion grade scale 40% No

50

Carry forward of passed components
No

Resit In Terminal Exam
Summer Yes - 1 Hour