Content of present website is being moved to . Registration of will be discontinued on 2020-08-14.
Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
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.
VI. Basic Math II.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Polar cones.


(Polar cone definition). For a nonempty set $C$ we define the polar cone $C^{\ast}$ : MATH

The following statement is a direct consequence of the definitions.


(Polar cone properties). For any nonempty set $C$ , we have

1. $C^{\ast}$ is a closed convex set.

2. MATH .

3. If $C\subset M$ for some set $M$ then MATH .


(Polar cone theorem). For any nonempty cone $C$ we have MATH

If $C$ is closed and convex then $C^{\ast\ast}=C$ .


First, we show that for any nonempty $C$ we have MATH . Indeed, by the definitions, for a fixed $x\in C$ MATH Therefore, MATH .

Next, we prove that for a closed nonempty $C$ , we have MATH .

Let $x\in C^{\ast\ast}$ . Since $C$ is closed, there exists the projection MATH . Let us translate the coordinate system so that MATH . Then by the proposition ( Projection theorem )-2 we have MATH Hence, MATH We already established that MATH Therefore, MATH but also MATH Hence, for a nonempty set $M\equiv C^{\ast}$ (empty $M$ is a trivial case) we have MATH By the definition of polar cone, we always have MATH for a nonempty $M$ . Hence, MATH .

Finally, we prove that MATH . By the proposition ( Polar cone properties ), we have MATH Therefore, MATH We already proved that MATH .

Notation. Index. Contents.

Copyright 2007