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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.
a. Affine sets and hyperplanes.
b. Convex sets and cones.
c. Convex functions and epigraphs.
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.

Affine sets and hyperplanes.


efinition

The set MATH is called "the line through $x$ and $y$ ". The set $M$ is called "affine" iff $l_{xy}\subseteq M$ for $\forall x,y\in M$ . The operation on set MATH is called "translation of $M$ by $a$ ". If $L$ is a subspace then the set $x+L$ is called "parallel to $L$ ".

Subspace is an affine set containing the origin. Every affine set is a translation of some subspace.

Definition

Dimension of the affine set is the dimension of the parallel subspace. Affine sets of dimension $n-1$ are called "hyperplanes". The set MATH is called "orthogonal complement of $M$ ."

Proposition

(Hyperplane representation)Hyperplanes are sets of the form MATH .

Proof

Subspaces of dimension $n-1$ are orthogonal complements of vectors. Hyperplanes are translations of such subspaces.

Proposition

Affine sets have the form MATH where $B$ is a matrix and $b$ is a vector. Consequently, affine sets are intersections of hyperplanes.

Proof

If $M$ is an affine set then $M=L+a$ for some subspace $L$ . Let MATH be the basis of $L^{\perp}$ then MATH . We set MATH as a union of columns and $b=Ba$ .

Intersection of affine sets is an affine set. Hence, we introduce the affine hull as follows.

Definition

(Affine hull). The affine hull of the set $S$ is MATH .





Notation. Index. Contents.


















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