In the following, the expected learning outcomes related to the joint courses Discrete Optimization and Nonlinear Optimization are labelled with, respectively, (1) and (2).

Knowledge and understanding

The students will learn:

- the key convex analysis concepts and results for optimization (1,2)

- a wide selection of optimization models arising in a variety of fields of application   (1)
- the main modeling techniques for Integer Linear Programming (ILP) problems  (1)
- the concepts of strong and ideal formulations, and how to stregthen a given formulation (1)
- some important relaxations, in particular the Lagrangian relaxation with the subgradient method (1)
- the cutting plane methods for generic and structured ILP problems  (1)
- the general scheme of the Banch-and-Cut method (1)
- the column generation method and Benders' deomposition for tackling large-scale problems (1)
- the greedy, local search and Tabu search heuristics (1)

- the optimality conditions for unconstrained nonlinear optimization  (2)
- the gradient, conjugate directions, Newton and quasi-Newton methods, with their convergence properties  (2)
- the subgradient method for convex non-smooth optimization  (2)
- the optimality conditions for constrained nonlinear optimization  (2)
- the concepts and results of Lagrangian duality  (2)
- the null-space and active set methods for quadratic programming problems  (2)
- the quadratic penalty method and the augmented Lagrangian method  (2)
- the logarithmic barrier method  (2)
- the fundamentals of quadratic sequential programming. (2)

Applying knowledge and understanding

The students will be able to:

- build optimization models starting from the decision problem descriptions, solve them using a mathematical modeling language and/or a state-of-the-art optimization solver  (1,2)

- enhance (linearize and strengthen) ILP formulations  (1)
- recognize an ideal ILP formulation based on the concept of totally unimodular matrices  (1)
- apply the studied methods, such as for instance cutting planes, column generation, Benders' decomposition and heuristics, to some small problems  (1)
- devise a Lagrangian relaxation for an assigned discrete optimization problem and discuss its strength  (1)

- establish whether a given nonlinear optimization problem is convex  (2)
- identify an appropriate method to tackle a given nonlinear optimization problem and apply it to small examples  (2)
- solve a constrained nonlinear optimization problem via the Karush-Kuhn-Tucker optimality conditions  (2)
- construct the Lagrangian dual of a given problem and establish the relationship with the primal problem  (2)
- solve (un)constrained nonlinear optimization problems with some of the studied methods in Matlab  (2)

- leverage on the acquired theoretical and algorithmic knowledge to answer relevant questions.  (1,2)

Communication skills

The students will be able to clearly describe any studied method, its main properties and its (dis)advantages with respect to other alternative algorithms.  (1,2)

The prerequisites for the course include linear programming (basic feasible solutions, simplex algorithm, duality), graph optimization (minimum spanning tree, shortest paths, maximum flow), Branch and Bound method for ILP and combinatorial optimization problems, and the basics of NP-completeness theory. These topics are covered in the course "Foundations of Operations Research".

A written exam covering all the material presented during the lectures, the sessions devoted to exercises, and the computer laboratory sessions and assignments. For the joint courses Discrete Optimization and Nonlinear Optimization, the material is that related to the lectures, exercises and computer laboratories of the corresponding part of the course, as detailed in the above program.

The written exam consists of several questions aimed at evaluating the level of (applying) knowledge and understanding of the main concepts, models, results and methods covered in the course as specified in the expected learning outcomes. In particular, it includes modeling questions, questions involving the application of the studied methods to assigned problems or the solution of numerical problems with appropriate algorithms, and theoretical questions related to the main concepts, methods and results. An open-answer question will also allow assessment of the communication skills.

A few brief computer laboratory tests scheduled at the end of some computer laboratory sessions.

The final grade is determined by adding to the grade of the written exam the points obtained in the computer laboratory tests.

The material will be made available during the semester