Advanced Optimization for Engineering Applications

The course presents the theory and practice of deterministic algorithms for locating the global solution of NP-hard optimization problems. Recurring themes and methods are convex relaxations, branch-and-bound, cutting planes, outer approximation and primal-dual approaches. Emphasis is placed on the connection between methods. As the course progresses, these methods will be applied and illustrated in the development of algorithms for mixed-integer linear programs, mixed-integer convex programs, nonconvex programs, and mixed-integer nonconvex programs. The broad range of engineering applications for these optimization formulations will also be emphasized.

Knowledge:

The course provides a presentation of the following topics:

• Introduction to the different classes of optimization (Linear Programming - LP, Mixed-Integer Linear Programming - MILP, Convex Nonlinear Programming, Mixed-Integer Convex Programming - MICP, Nonconvex Nonlinear Programming - NLP, and Nonconvex Mixed-Integer Programming - MINLP
• Motivation for Global Optimization
• Background concepts: Convex Optimization, Optimality Criteria for NLPs, Duality Theory and Computational Complexity
• Attaining Global Information: Interval Analysis, Convex and Concave Relaxations
• MILPs - Problem formulation and Branch-and Bound based solution algorithms
• NLPs - Solution using spatial Branch-and-Bound algorithms
• MICPs - Solution using Outer Approximation and Generalized Benders Decomposition
• MINLPs - Solution using Branch-and-Bound based algorithms, Nonconvex Generalized Benders Decomposition and Nonconvex Outer Approximation

Skills:

• Have an understanding of how to formulate and solve optimization problems appropriately using commercial software (e.g. JuMP in Julia, or GAMS)
• Have an overview of the workings behind several state-of-the-art algorithms for various problem classes. Know which algorithms are appropriate for what types of problems

General Competence:

• Learning how to use computational tools to support decision-making for a variety of engineering problems

A hybrid teaching form will be used, where a considerable degree of Self-learning is combined with Q&A Colloquiums with one or more lecturers and the students of this course. As mentioned below, videos from a similar course at MIT will also be made available together with a text book and a considerable set of slides explaining some of the details of the curriculum. The course will also have assignments with solutions.

The course is given every two years, starting autumn 2025.