### Continuous Optimization - M1 - 6EC

Continuous Optimization [Fall 2019]

Prerequisite knowledge

The student should have a solid bachelor level knowledge linear algebra and multivariate analysis. The student should also have knowledge of linear optimization 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

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 optimization 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 optimization problems.

Learning goals

The student will be able to:

• Prove results on (convex) optimization problems.
• Solve the KKT conditions for basic constrained optimization 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 optimization problems.
• Use a range of techniques to solve both unconstrained and constrained continuous optimization problems, and prove results on these techniques.
• Formulate and recognise conic optimization problems, along with being able to construct their dual problems.

Lecturer