Prerequisite/Recommended prerequisite for participation in the module

Content, progress and pedagogy of the module

Engineering systems and design problems can often be compactly described analyzed 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

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

Skills

• Identify optimization problems and cast them into standard form
• Identify types of extreme (minima, maxima, local, global, etc.)
• Apply eigenvalue and singular value decomposition to relevant matrix problems
• Have understanding of state space descriptions of systems of linear differential equations
• Apply numerically robust methods to solve systems of equations
• Apply and implement the following numerical optimization methods to unconstrained optimization problems: Steepest Descent, Newton's method, Gauss-Newton method
• Apply and interpret least-squares in solving over-determined systems of equations
• Apply the Lagrange multiplier method to constrained convex optimization problems

Competences

• Apply linear algebra theory to analyze engineering systems in their field
• State and analyze engineering design problems in their field as systems of equations or standard optimization problems
• Select the appropriate matrix factorization or numerical optimization method to solve engineering design problems in their field

Type of instruction

Lectures with exercises. Student projects on engineering application in their field

Exam

Exams