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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.

Conjugate duality.


For any convex function $f$ MATH


Each affine function MATH corresponds to a hyperplane. By the proposition ( Supporting hyperplane theorem ), for any point below epigraph of $f$ there exists a hyperplane that separates such point from the $epi~f$ . Hence, for any such point MATH MATH there exists a pair MATH s.t. MATH and MATH for $\forall y\in dom~f$ .

Duality picture 1
$epi~f$ is the intersection of the upper half planes.


The set $epi~f$ is equal to intersection of the upper half-planes defined by the hyperplanes from proof of the previous statement.

Let us denote $F=epi~f$ . Let us introduce a set MATH Such set is not empty and it is an epigraph of some function because if MATH then MATH for all $\beta>\beta_{0}$ . Let us denote such function $f^{\ast}$ . By definition MATH

Duality picture 2
Geometrical meaning of $f(x^{\ast})$ .


Let us introduce MATH MATH MATH Observe that MATH . Indeed, MATH means that $\exists c=const$ MATH or MATH where the $x^{\ast}$ and MATH run through all the hyperplanes that define $epi~f$ . For the same reason the infimum of such $c$ is the MATH : MATH


(Conjugate duality theorem). We define the operation of taking a dual function $f^{\ast}$ by MATH Then MATH for all proper convex functions.

The closure part: $cl\ f$ comes from the fact that taking affine envelopes includes boundary points of the $epi~f$ into the final result $epi~f^{\ast \ast}$ .

a. Support function.
b. Infimal convolution.

Notation. Index. Contents.

Copyright 2007