Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
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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.
a. Support function.
b. Infimal convolution.
VI. Basic Math II.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Infimal convolution.

he following statement is direct consequence of definitions.


(Infimal convex function)Let MATH be a convex set. The function MATH is convex.


(Infimal convolution) The function MATH is called infimal convolution.


If MATH are convex functions then MATH is a convex function.


Apply the result ( Infimal convex function ) to the set MATH .


(Duality of infimal convolution and addition). Let MATH be convex functions. Then MATH


By definition of infimal convolution ( Infimal convolution ) and the duality operation ( Conjugate duality theorem ) we have MATH


(Infimal convolution of support functions). Let MATH be non-empty convex sets, MATH . Then MATH

Notation. Index. Contents.

Copyright 2007