CALCULUS AND OPTIMIZATION

This course (C&O) represents a foundational course in the curriculum of every master's student. C&O aims to contribute to training professional figures suitable for employment in industry and research, in fields where high-level training is required in the areas of software development and design, artificial intelligence, and cybersecurity. In this regard, C&O covers the main paradigms of computing and mathematical modeling, applied in order to enable students to acquire practical skills for the job market, technology transfer, and applied research. For the degree in COMPUTER SCIENCE AND INFORMATION TECHNOLOGY, the main objectives of C&O can be summarized more specifically as follows:

(a) to provide basic knowledge of Integral-Differential Calculus in R^n, highlighting and distinguishing the use of first and second-order information associated with the used functions;

(b) to develop the ability to create, solve, and analyze a Mathematical Programming (Optimization) model for the current problem. The course offers contents for modeling real problems, even in the presence of technological constraints. This approach specifically aims to develop the analysis and synthesis skills related to real problems in presence of scarce resources.

Virtually every quantitative problem that students met in previous courses can be studied using tools developed in C&O, allowing students to simultaneously develop an analytical approach, but in view of managing problems related to applications and technology transfer.

The active involvement of students to the activities of the course, including individual study, will allow to pursue the following results:

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.

Students should provably know the contents of the MATH course. In particular, students must be able to work with the following
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.

The course will cover the following topics:

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)

The next references are advised to better assimilate the contents of the course. Apart from the handouts by the teacher, the other book-references can be considered "not essential":

(***) 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.

A written test is assigned, lasting about 1.5-2.5 hours. Then, based on the results of its correction, the teacher communicates the students if he/she is allowed to join the oral part of the exam, which takes place in the same day of the written part. There will be also 1 intermediate assignment during the period of lessons. The intermediate assignment will have a grade in the range 0-30 / 30, and will focus on issues in reference (***). The teacher publishes fully solved exam exercises and intermediate calls, relative to the previous academic years, on https://moodle.unive.it

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.

written and oral

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 subject;
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 subject;
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
subject;
D. Honors will be granted when there is excellent knowledge and applied understanding in reference to the
curriculum, exceptional judgment, and communication skills.

This is a "Face-to-face" course which adopts also additional teaching materials available on the e-learning platform https://moodle.unive.it/ .
The online teaching materials report the contents of both lessons and exercises. Students are required to actively participate, practice and do the proposed exercises.

See also the e-learning platform https://moodle.unive.it for further info/documents/downloads.

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.

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