# CALCULUS AND OPTIMIZATION

2022/2023
Official course title
CALCULUS AND OPTIMIZATION
Course code
CM0469 (AF:398283 AR:214931)
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
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The main target of the course is that of providing basic notions of Calculus and Mathematical Programming (Optimization), which will be used in several other courses.
This course (C&O) represents a basic subject in the curriculum of a Master Degree program in INFORMATICA - COMPUTER SCIENCE. 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.
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, calculus with one-two unknowns) as a Prerequisite.
Students should provably know the contents of the MATHEMATICS 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 knowledge expected 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
3. Derivatives and Directional Derivatives
4. Differentiation and the Chain Rule
5. The Taylor 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)
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 at 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 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 assignments during the period of lessons.
This is a "Face-to-face" / "Blended" (depending on the evolution of CoVID-19 emergence) 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.
English