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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.

Convex functions and epigraphs.


efinition

(Convex and proper function). The "epigraph" of a function $f$ is the set MATH , see the picture ( Picture of convex function ). The function $f$ is "convex" iff the set $epi(f)$ is convex. The "effective domain" is the set MATH .

The function is "proper" if the epigraph is nonempty and does not contain a vertical line.


Convex function picture
Convex function $f$ acting from $\U{211d} ^{2}$ to $\U{211d} $ . Level sets $lev_{\alpha}(f)$ .

The consideration of this entire chapter on convex analysis is restricted to proper functions. Hence, all functions that are said to be convex are also presumed to be proper.

Proposition

(Main property of convex function). A function $f$ is convex iff MATH .

Proposition

A smooth function $f$ is convex iff the matrix of second derivatives is non-negatively determined.

Proof

Fix two points $x_{0}$ and $x_{1}$ and denote MATH Let $D$ be the matrix of second derivatives MATH taken at the point $x_{0}$ . Note that MATH We use the Taylor decomposition MATH If we assume that the function is convex then we have, by the proposition ( Main property of convex function ), MATH Hence, MATH for MATH . Therefore, MATH for any $y=x_{1}-x_{0}$ .

Proposition

(Preservation of convexity).

1. If MATH are convex functions and MATH are positive real numbers then MATH is convex.

2. If MATH is a convex function and $A$ is a matrix then MATH is convex.

3. If MATH are convex functions and $I$ is an arbitrary index set then MATH is convex.





Notation. Index. Contents.


















Copyright 2007