Optimisation IDX S5
Module aims
The aim of this module is to equip you with the tools to formulate and solve general constrained and unconstrained optimisation problems. The module covers several introductory topics in optimisation such as necessary and sufficient conditions of optimality, basic optimization algorithms (gradient, Newton, conjugate directions, quasi-Newton), Kuhn-Tucker conditions, penalty method, recursive quadratic programming, and global optimization. Each topic is covered in a mathematical rigorous way with attention to regularity, convergence conditions, and complexity.
The module assumes prior basic calculus and linear algebra knowledge such as multivariable calculus, sequences, compactness, and eigenvalues.
Learning outcomes
Upon successful completion of this module, you will be able to:
1 - Formulate simple unconstrained and constrained optimization problems
2 - Classify optimal solutions
3 - Apply the correct methods to solve such problems
4 - Write basic unconstrained optimization algorithms and assess their convergence and numerical properties
5 - Apply the notion of penalty in the solution of constrained optimization problems
6 - Change constrained optimization problems into equivalent unconstrained problems
7 - Apply basic algorithms for the solutions of global optimization problems
Module syllabus
Necessary and sufficient conditions of optimality
Line search
The gradient method, Newton's method, conjugate direction methods, quasi-Newton methods, methods without derivatives
Kuhn-Tucker conditions
Penalty function methods
Exact methods
Recursive quadratic programming
Global optimization
Teaching methods
This module is taught as a traditional module with theoretical lectures supported by exercises and examples.
The lectures cover the topics from a theoretical point of view. Concepts, problems and solution methods are explained and justified from a theoretical and intuitive point of view. The coursework is an application of some of the methods seen in the module to a well-known optimization problem.
The module makes use of a set of lecture notes which contains all material taught in the lectures and over 100 exercises.
Assessments
A final exam (3hrs written examination in the Summer term) will test the theoretical concepts covered during the lectures (75%).
Formative feedback will be provided on an ongoing basis: in the classroom (as general comments) and during office hours.
Module leaders
Professor Alessandro Astolfi
