CALCULUS AND OPTIMIZATION
- Academic year
- 2024/2025 Syllabus of previous years
- Official course title
- CALCULUS AND OPTIMIZATION
- Course code
- CM0469 (AF:513722 AR:286754)
- Modality
- On campus classes
- ECTS credits
- 6
- Degree level
- Master's Degree Programme (DM270)
- Educational sector code
- MAT/09
- Period
- 1st Semester
- Course year
- 1
- Where
- VENEZIA
- Moodle
- Go to Moodle page
Contribution of the course to the overall degree programme goals
This course (C&O) represents a basic subject in the curriculum of a Master Degree program in COMPUTER SCIENCE AND INFORMATION TECHNOLOGY. It aims at providing:
(a) basic notions of Calculus (including the analysis of curves-surfaces in R^n);
(b) the capability to develop, create, analyze and solve a Mathematical Programming (Optimization) model for the problem in hand.
Virtually all the models in students previous courses can gain advantage of the techniques studied in C&O.
Expected learning outcomes
1) Knowledge and Understanding: of basic and advanced tools relative to Calculus, involving 'n' real variables;
2) Capability to Apply Knowledge and Understanding: to generate/manipulate quantitative models of Calculus, with reference to all applied sciences;
3) Capability to Judge and Interpret: using and manipulating mathematical models, on the basis of specific and analytical indicators.
The course requires a basic knowledge of Math (numbers, sequences, linear algebra, matrix algebra, calculus with one-two unknowns) as a Prerequisite.
Pre-requirements
concepts: systems of equalities and inequalities, linear algebra for matrices, extreme points of functions with one unknown, functions with
two unknowns, derivatives and integrals of functions with one unknown.
A (possible) initial test will indicate the level of expected knowledge by the attendees.
A (possible) final test will indicate the acquired level of knowledge by the attendees.
Contents
1. Generalities on functions in R^n, Tangential and Normal vectors
2. Eigenvalues and Eigenvectors of matrices
3. Derivatives and Directional Derivatives
4. Differentiation and the Chain Rule
5. The Taylor's expansion
6. Implicit Function Theorem (Dini's Theorem)
7. Fubini’s Theorem (notes)
8. Exact differentials, Multiple Integration and the role of the Jacobian (notes)
10. Stokes’ Theorem (notes)
11. Local/Global Minima/Maxima for functions with 'n' unknowns
12. Karush-Kuhn-Tucker and Constraint Qualification conditions
13. Convexity and optimality conditions (necessary/sufficient conditions)
14. Mean Value Theorems
15. Optimization methods for unconstrained/constrained problems (introduction)
16. Gradient methods and Projected Gradient methods
17. Linesearch procedures
18. Conjugate Gradient methods and Quasi Newton methods
19. Active set methods (notes)
20. Penalty/Barrier methods (notes)
21 Lagrangian and Augmented Lagrangian methods (notes)
Referral texts
(***) Afternotes by the teacher, available on the platform https://moodle.unive.it/
M.S.Bazaraa, H.D.Sherali, C.M.Shetty (1993) "Nonlinear Programming - Theory and Algorithms (2nd edition), John Wiley & Sons.
D.P.Bertsekas (1982) "Constrained Optimization and Lagrange Multiplier Methods", Academic Press.
D.P.Bertsekas (1995) "Nonlinear Programming", Athena Scientific, Belmont, Massachusetts, USA.
R.Walter (1976) "Principles of Mathematical Analysis", McGraw-Hill.
C.H.Edwards, “Advanced Calculus of Several Variables”, Dover Publications, 2003
B.T.M. Apostol “Calculus: Multivariable Calculus and Linear Algebra, with Applications to Differential Equations and Probability, vol. II, Second Edition”, John Wiley and Sons, Inc., 1973
J.Nocedal, S.J.Wright, “Numerical Optimization, Second Edition”, Springer, 2006.
S.Boyd, L.Vandenberghe “Convex Optimization”, Cambridge University Press, 2009.
Assessment methods
For students who have passed the intermediate call with a grade >= 18, the exam in each call will include:
- Written Part with 2/3 exercises + 1/2 written questions (on the entire programme excluding the reference (***));
- (Possibly) Oral Part in case the evaluation of the written part + intermediate assignment were not sufficient.
For students who have NOT joined the intermediate call or have NOT passed it with a grade >= 18, the exam in each call with include:
- Written Part with 4/5 exercises + 2/3 written questions (on the entire programme);
- (Possibly) Oral Part in case the evaluation of the written part were not sufficient.
The grade 'X' will be given by the (arithmetic) mean with rounding of the grades reported in the intermediate call (where present) and in the written part. The Oral Part may last 20-25 minutes and will include questions on the written part and on the programme. Moreover, the oral part may contribute to increase/decrease the grade 'X' of at most 3/30.
In any case, regardless of attending or non-attending mode:
A. Scores in the range of 18-22 will be assigned when:
- Adequate knowledge and applied understanding in reference to the curriculum are demonstrated.
- Limited ability to collect and/or interpret data and formulate independent judgments.
- Sufficient communication skills, especially regarding the use of specific language related to the economic
functioning of public services.
B. Scores in the range of 23-26 will be assigned when:
- Good knowledge and applied understanding in reference to the curriculum are demonstrated.
- Fair ability to collect and/or interpret data and formulate independent judgments.
- Adequate communication skills, especially regarding the use of specific language related to the economic
functioning of public services.
C. Scores in the range of 27-30 will be assigned when:
- Good or excellent knowledge and applied understanding in reference to the curriculum are demonstrated.
- Reasonable or excellent ability to collect and/or interpret data and formulate independent judgments.
- Fully appropriate communication skills, especially regarding the use of specific language related to the
economic functioning of public services.
D. Honors will be granted when there is excellent knowledge and applied understanding in reference to the
curriculum, exceptional judgment, and communication skills.
Teaching methods
The online teaching materials report the contents of both lessons and exercises. Students are required to actively participate, practice and do the proposed exercises.
Teaching language
Further information
Accessibility, Disability and Inclusion
Accommodation and support services for students with disabilities and students with specific learning impairments
Ca’ Foscari abides by Italian Law (Law 17/1999; Law 170/2010) regarding support services and accommodation available to students with disabilities. This includes students with mobility, visual, hearing and other disabilities (Law 17/1999), and specific learning impairments (Law 170/2010). If you have a disability or impairment that requires accommodations (i.e., alternate testing, readers, note takers or interpreters) please contact the Disability and Accessibility Offices in Student Services: disabilita@unive.it.
Type of exam
2030 Agenda for Sustainable Development Goals
This subject deals with topics related to the macro-area "Human capital, health, education" and contributes to the achievement of one or more goals of U. N. Agenda for Sustainable Development