Optimization models and methods [ LINMA2471 ]

Course documents (notes, slides, exercises and homeworks) are available on Moodle.

https://moodleucl.uclouvain.be/course/view.php?id=8194

A basic optimization course (such as LINMA1702) and basic knowledge in real analysis and linear algebra (such as provided by FSAB1101 and FSAB1102)

Linear optimization, convex optimization (including structured conic optimization) ; duality and applications ; interior-point methods ; first-order methods ; trust-region methods ; use of a modeling language.

Learning outcomes:
AA1.1, AA1.2, AA1.3
AA2.1, AA2.2, AA2.4, AA2.5
AA5.3, AA5.5

More specifically, at the end of the course the student will be able to :

• recognize the possiblity of formulating or converting a problem into a linear, convex or conic optimization program
• exploit the concept of duality in order to understand a problem, produce optimality or impossibility certificates, carry out sensitivity analysis or formulate robust problems
• describe, analyze and implement advanced algorithms to solve linear, convex or non-linear optimization problems
• use a modeling language to formulate and solve optimization problems, while understanding and exploiting the formal separation between model, data and resolution algorithm

Transversal learning outcomes :

• use a numerical/computational software tool such as MATLAB, or a modeling language such as AMPL
• formulate, analyze and solve optimization models, in a small group
• write a report about the formulation, analysis and resolution of optimization models, in a small group

The contribution of this Teaching Unit to the development and command of the skills and learning outcomes of the programme(s) can be accessed at the end of this sheet, in the section entitled “Programmes/courses offering this Teaching Unit”.

Students will be evaluated with an individual written exam, based on the above-mentioned objectives. Students also carry out a series of homeworks in small groups (also taken into account for the final grade).

The course is comprised of lectures, exercise sessions and computer labs (for the AMPL modeling language).

Models: Advanced modeling techniques for linear and convex optimization ; theorem of the alternative, linear and convex duality ; sensitivity analysis and robust optimization ; conic optimization (linear, conic quadratic and semidefinite programming) ; Lagrangian duality
Methods: interior-point methods for linear optimization (short- and long-step path-following methods) and for convex optimization (self-concordant barriers), first-order methods for convex optimization, algorithmic complexity ; trust-region methods ; introduction to the AMPL modeling language.
Applications in various domains, such as data analysis, machine learning, finance, shape or structural  optimization (mechanics), telecommunications, etc.

• Convex Optimization, Stephen Boyd et Lieven Vandenberghe, Cambridge University Press, 2004.
• Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications, Aharon Ben-Tal, Arkadi Nemirovski, SIAM 2001.
• Interior point methods for linear optimization, Cornelis Roos, Tamas Terlaky, Jean-Philippe Vial, Springer, 2006.
• Introductory Lectures on Convex Optimization: A Basic Course, Yurii Nesterov, Kluwer, 2004.
• Trust-region methods, A. Andrew R. Conn, Nicholas I. M. Gould, Ph. Philippe L. Toint, SIAM, 2000.