Content of present website is being moved to . Registration of will be discontinued on 2020-08-14.
Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Printable PDF file
I. Basic math.
II. Pricing and Hedging.
III. Explicit techniques.
IV. Data Analysis.
V. Implementation tools.
1. Finite differences.
2. Gauss-Hermite Integration.
3. Asymptotic expansions.
4. Monte-Carlo.
5. Convex Analysis.
A. Basic concepts of convex analysis.
B. Caratheodory's theorem.
C. Relative interior.
D. Recession cone.
E. Intersection of nested convex sets.
F. Preservation of closeness under linear transformation.
G. Weierstrass Theorem.
H. Local minima of convex function.
I. Projection on convex set.
J. Existence of solution of convex optimization problem.
K. Partial minimization of convex functions.
L. Hyperplanes and separation.
M. Nonvertical separation.
N. Minimal common and maximal crossing points.
O. Minimax theory.
P. Saddle point theory.
Q. Polar cones.
R. Polyhedral cones.
S. Extreme points.
T. Directional derivative and subdifferential.
U. Feasible direction cone, tangent cone and normal cone.
V. Optimality conditions.
W. Lagrange multipliers for equality constraints.
X. Fritz John optimality conditions.
Y. Pseudonormality.
Z. Lagrangian duality.
[. Conjugate duality.
VI. Basic Math II.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Caratheodory's theorem.


Let $X$ be a non-empty subset of $\QTR{cal}{R}^{n}$ .

1. For every MATH there are linearly independent vectors MATH $x_{i}\in X$ , $i=1,...,m$ such that MATH for some MATH and finite $m>0$ .

2. For every MATH there are vectors MATH , $i=1,...,m$ such that MATH for some MATH and the vectors MATH are linearly independent.


The Caratheodory theorem does not state that MATH might serve as a fixed basis. Indeed, on the picture ( Caratheodory theorem remark ) if the set X is open then for any pair of vectors $x_{1}$ and $x_{2}$ from X a point MATH may be found outside of the area span by the positively linear combinations of $x_{1}$ and $x_{2}$ .

Caratheodory theorem remark figure
Caratheodory theorem remark.


1. The definition of MATH provides that there are some vectors MATH such that MATH If such vectors are linearly dependent then there are numbers $\beta_{i}$ MATH We take a linear combination of two equalities MATH and note that for at least one $i$ the $\beta_{i}$ is positive. Hence, a $\gamma$ exists such that all MATH are non negative and MATH for at least one index $i_{0}$ . Hence, we decreased the number of terms in the sum. We continue this process until MATH are linearly independent.


2. The definition of MATH provides that there are some vectors MATH such that MATH We consider MATH and restate the above conditions as MATH Therefore, MATH . The first part of the theorem applies and the vectors MATH may be assumed linearly independent. Hence, no all non-zero MATH exist such that MATH Equivalently, MATH We express the $\beta_{1}$ from the second equation and substitute it into the first. We obtain the following consequence MATH We conclude that no all non-zero MATH exist such that the above is true. Hence, the MATH are linearly independent.

Notation. Index. Contents.

Copyright 2007