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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
Author
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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.

Relative interior.


roposition

Let C be a nonempty convex set consisting of more then one point.

a. ( Non emptiness of relative interior). The $ri\left( C\right) $ is not empty and MATH . If MATH then there are vectors MATH such that MATH span the subspace parallel to MATH .

b. ( Line segment principle). If MATH and MATH then all point of the line connecting $x$ and $\bar{x}$ , except possibly the $\bar{x}$ , belong to $ri\left( C\right) $ .

Let $a,$ $a\in C$ be a vector such that MATH is a subspace. One can choose vectors MATH such that $y_{k}\in C-a$ , $y_{k}$ are linearly independent and the linear span of MATH is MATH . All convex combinations of MATH belong to $C$ and also belong to $ri\left( C\right) $ . Hence, the $ri\left( C\right) $ is not empty. We construct MATH as claimed in (a) by taking $x_{k}=y_{k}+a$ . Consequently, MATH .

The statement (b) is evident from the picture ( Relative interior ).


Relative interior figure
Relative interior.





Notation. Index. Contents.


















Copyright 2007