References:

[1] Zorich V. A. - Mathematical Analysis I

[2] Zorich V. A. - Mathematical Analysis II

[3] Boyd S., Vandenberghe L. - Convex Optimization

[4] Netuka I. - The Change-of-Variable Theorem for the Lebesgue Integral

Exercise Sheets:

[Sheet 1] [Sheet 2] [Sheet 3] [Sheet 4] [Sheet 5] [Sheet 6]

Classroom diary:

1 - 19/03 - Review of normed multilinear maps. The differential of a function between normed vector spaces. Uniqueness of the differential. Differentiability implies continuity. General properties of the differential. Partial derivatives. Ref. [2, Chapters 10.2, 10.3]

2 - 26/03 - Examples of differentials and the geometric meaning of the gradient. Regular curves and their arc-length. Equivalent curves. An example of curve with infinite length. Curvilinear integrals (of the first kind). Ref. [1, Chapters 6.4.2]

3 - 09/04 - The finite increment theorem (opposed to the one-dimensional mean value theorem). Some consequences: sufficient condition for the differentiability, Lipschitz property. Directional derivatives and their relation with the differential. Higher order derivatives and how to calculate them. Ref. [2, Chapters 10.4, 10.5.1, 10.5.2]

4 - 11/04 - Symmetry of higher order derivatives. Review of bilinear forms and the Hessian matrix. Differential of multilinear functions. Ref. [2, Chapters 10.5.3, 10.5.4]

â€‹

5 - 16/04 - Taylor polynomial. Study of the Extrema of functions. Some examples. The fundamental lemma of the calculus of variations. Ref. [2, Chapter 10.6]

6 - 23/04 - Euler-Lagrange equation. Newton's second law of motion as solution of a variational principle. Review on convex functions. Setting for the unconstrained convex optimization. Ref. [2, Chapter 10.6] [3, Chapters 3, 9.1]

7 - 30/04 - The "strongly convex hypothesis". Relations between the norm of the gradient and the quality of the approximations of the minimum. Generalities on descent algorithms: "descent direction" and "ray search". Two types of ray search: "exact" and "backtracking". Gradient descent. Ref. [3, Chapters 9.1, 9.2]

8 - 07/05 - Convergence analysis of the gradient descent. The fixed point theorem on complete metric spaces. Statement and explanation of the implicit function theorem for normed spaces. Ref. [3, Chapters 3, 9.3.1] [2, Chapter 9.7] [2, Chapter 10.7]

9 - 14/05 - Proof of the implicit function theorem. Refinements of the implicit function theorem for differentiable functions (no proof). The inverse function theorem in R^m. Some applications: polar coordinates, local rectification of curves. Ref. [2, Chapter 10.7] [1, Chapter 8.6.1]

10 - 21/05 - Submanifolds of R^m. Submanifolds given by "a system of functions of maximal rank". The tangent space. Extrema with constraints: generalities. Necessary condition for a constrained extremum in terms of tangent spaces. [1, Chapter 8.7.1 -> 8.7.3]

11 - 28/05 - The Lagrange function. Sufficient condition for a constrained extremum in the case of system of functions (no proof). Examples of constrained optimization. Local decomposition of a diffeomorphisms into elementary ones. [1, Chapter 8.7.1 -> 8.7.3, 8.6.4]

12 - 04/06 - Complements of measure theory. Change of variables for the Lebesgue integral (complete proof and examples) [4]

13 - 11/06 - Exercises (some of them from the Sheet 6).

â€‹