OPTIMIZATION

The course aims at presenting the basic concepts and methods of optimization. The course provides students with the analytic tools to model and foresee situations in which a single decision-maker has to find the best choice. The course focuses on applications in economics, engineering, and computer science. At the end of the course, the students will be able to formulate mathematically real-life problems, solve them by applying numerical methods, and realize what the optimal choice is.

The goals of the course are:

Knowledge and understanding: to acquire base knowledge that allows students to study optimization problems and apply opportune techniques to solve decision-making problems. The students will be able to use algorithms for both linear and nonlinear programming problems.
Applying knowledge and understanding: to identify and model real-life decision-making problems. In addition, through real examples, the student will be able to find correct solutions for complex problems.
Making judgments: to choose and solve autonomously complex decision-making problems and to interpret the solutions.
Communication skills: to acquire base communication and reading skills using technical language.
Learning skills: to provide students with theoretical and practical methodologies and skills to deal with optimization problems, ranging from computer science to engineering; to acquire further knowledge on the problems related to applied mathematics.

Teaching Organization

credit value 6 - 48 hours

total study 150 hours

102 hours of individual study
24 hours of frontal lecture

24 hours of exercises

 

For this course, there will be 2 hours of teaching per lecture twice a week. During the classroom lessons, a graphics tablet will be used. The hand-written slides will be available on the platform Studium. For each topic, exercises will be solved by the teacher or proposed to students.

Should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, in line with the programme planned and outlined in the syllabus.

Information for students with disabilities and/or SLD

To guarantee equal opportunities, in compliance with the laws in force, interested students can ask for a personal interview in order to plan any compensatory and/or dispensatory measures, based on the didactic objectives and specific needs. It is also possible to contact the referent teacher CInAP (Center for Active and Participated Integration - Services for Disabilities and/or SLD) of our Department.

No requirements.

Course attendance is strongly recommended.

1. LINEAR PROGRAMMING (12 hours)

Primal simplex method. Duality and dual simplex method.

2. INTEGER PROGRAMMING (6 hours)

Integer programming models. Transportation and assignment problems. Branch and Bound method and cutting plane methods in integer programming; KP problem; TSP problem.

3. NONLINEAR PROGRAMMING (6 hours)

Optimality conditions for nonlinear problems; numerical methods for constrained and not constrained problems.

4. SOLVING OPTIMIZATION PROBLEMS (24 hours)

Analitycal solutions. Use of software (GeoGebra, Excel, Matlab, Mathematica).

[1]   F.S. Hillier, G.J. Lieberman, Introduction to Operations Research, Mc Graw Hill, 2020
[2]  O.L. Mangasarian, Nonlinear Programming, SIAM Classics in Applied Mathematics.
[4]   D.P. Bertsekas, Nonlinear Programming, Athena Scientific.

Teaching material will be given during the course.

The final exam consists of an oral test during which the candidate is also requested to solve a numerical exercise. The final grade is established on the basis of the answers given by the candidate and the solving of the numerical example.

It is possible to carry out one or more ongoing tests on the contents of the course. Passing the ongoing tests exempts you from the final interview.

Final grades will be assigned taking into account the following criteria:

Rejected: Basic knowledge has not been acquired. The student is not able to solve simple exercises.

18-23: Basic knowledge has been acquired. The student solves simple exercises, has sufficient communication skills, and makes judgements.

24-27: All the knowledge has been acquired. The student solves all the proposed exercises making few errors and has good communication skills and making judgements.

28-30 cum laude: All the knowledge has been completely acquired. The student applies knowledge and has excellent communication skills, learning skills and making judgements.

 

Learning assessment may also be carried out online if the conditions should require it.

 

Simplex method.  Linear programming examples. Integer linear programming and use of branch and bound method. Knapsack problem. Optimality conditions in linear programming. Optimality conditions for non linear problems. KKT conditions. Penalty and barrier methods.