Content of present website is being moved to www.lukoe.com/finance . Registration of www.opentradingsystem.com will be discontinued on 2020-08-14.
Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
Author
Printable PDF file
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.

Support function.


efinition

The indicator function MATH of a convex set $C$ is a function of the form MATH The support function MATH is the conjugate of the indicator function.

According to the definition

MATH (Support function)

Note that MATH is a positively homogenous function of $x^{\ast}$ . Suppose $f\left( x\right) $ is some proper convex positively homogenous function. Consider the conjugate MATH By the positive homogeneity MATH . Consequently, for any $\lambda>0$ , MATH Hence, MATH is either 0 or $+\infty$ . Introduce the set MATH Such set is a $dom~f^{\ast}$ . Indeed, if MATH then MATH and 0 is reached by scaling with $\lambda$ . Then MATH . On the other hand if MATH then MATH and $+\infty$ is reached by scaling.

Summary

(Convex homogenous function property). If $f\left( x\right) $ is a proper convex positively homogenous function then MATH and MATH

The last part of the summary follows from the result ( Conjugate duality theorem ). We established one-to-one correspondence between convex sets and proper convex positively homogenous functions.





Notation. Index. Contents.


















Copyright 2007