Prerequisite/Recommended prerequisite for participation in the module

Content, progress and pedagogy of the module

Purpose
Engineering systems and design problems can often be compactly described analysed and manipulated using matrices and vectors. Moreover, tractable solutions to design problems can be obtained by casting the design problems as optimization problems. For the class of linear and quadratic problems, the solutions can be obtained by solving systems of equations. In computer programs, this is achieved via matrix factorizations. For the larger class of convex problems, no closed-form solution may exist and numerical methods must be applied. This course aims at teaching numerically robust methods for solving systems of equations and, more generally, convex optimization problems, including also standard constrained problems.

Learning objectives

Knowledge

• Must have knowledge about convex functions and sets, norms, special matrices
• Must have understanding of how to classify and solve systems of equations and convex optimization problems
• Must have understanding of numerical aspects of solving systems of equations and convex optimization problems
• Must have knowledge about Lagrange multipliers
• Must have understanding of matrix factorizations and their properties

Skills

• Must be able to identify optimization problems and cast them into standard form
• Must be able to identify types of extrema (minima, maxima, local, global, etc.)
• Must be able to apply Eigen value and singular value decomposition to relevant matrix problems
• Must have understanding of state space descriptions of systems of linear differential equations
• Shall be able to apply numerically robust methods to solve systems of equations
• Shall be able to apply and implement the following numerical optimization methods to unconstrained optimization problems: Steepest Descent, Newton's method, Gauss-Newton method
• Shall be able to apply and interpret least-squares solutions when solving over-determined systems of equations
• Shall be able to apply the Lagrange multiplier method to constrained convex optimization problems

Competences

• Are able to apply linear algebra theory to analyse engineering systems in their field
• Can state and analyse engineering design problems in their field as systems of equations or standard optimization problems
• Are able to select appropriate matrix factorization or numerical optimization methods to solve engineering design problems in their field

Type of instruction

The programme is based on a combination of academic, problem-oriented and interdisciplinary approaches and organised based on the following work and evaluation methods that combine skills and reflection:

• Lectures
• Classroom instruction
• Project work
• Workshops
• Exercises (individually and in groups)
• Teacher feedback
• Reflection
• Portfolio work

Extent and expected workload

Since it is a 5 ECTS course module, the work load is expected to be 150 hours for the student

Exam

Exams