Aim of the course

This course aims to provide a concise introduction into the basics of continuous unconstrained, constrained and conic optimization.

The course starts with an analysis of convex sets and convex functions. Duality in convex optimisation is the next topic. We consider Lagrange- and saddle-point duality. Then an introduction into theory and basic algorithms for unconstrained and constrained nonlinear problems is presented. The courses finished with the study of conic optimisation problems.

Learning goals

The student will be able to:

• Prove results on (convex) optimisation problems.
• Solve the KKT conditions for basic constrained optimisation problems.
• Be able to formulate the Lagrange and Wolfe Dual, and understand/prove basic results on these problems.
• Give both sufficient and necessary optimality conditions for constrained continuous optimisation problems.
• Use a range of techniques to solve both unconstrained and constrained continuous optimisation problems, and prove results on these techniques.
• Formulate and recognise conic optimisation problems, along with being able to construct their dual problems.

Assessment

The course will be assessed by a final exam.

Prerequisites

The student should have a solid bachelor level knowledge linear algebra and multivariate analysis. The student should also have knowledge of linear optimisation and convex analysis to the level of being able to follow the text and do the exercises from the following:

• Linear Programming, A Concise Introduction, Thomas S. Ferguson:

Available at https://www.math.ucla.edu/~tom/LP.pdf

Chapters 1 and 2, along with the accompanying exercises.

• Convex Optimization, Stephen Boyd and Lieven Vandenberghe:

Available at http://stanford.edu/~boyd/cvxbook/

Sections: 2.1, 2.2 and 3.1.

Exercises (from the book): 2.1, 2.2, 2.12, 3.1, 3.3, 3.5 and 3.7

Literature

Lecture slides will be provided online during the course.

Lecturer

Krzysztof Postek, Erasmus University Rotterdam, e-mail: postek@ese.eur.nl