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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.
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.

Projection on convex set.


roposition

(Projection theorem). Let $C$ be a nonempty closed convex set.

1. For any MATH there exists a unique vector MATH called the projection of $x$ on $C$ .

2. The MATH could be defined as the only vector with the property MATH If the $C$ is affine and $S$ is a subspace parallel to $C$ then the above may be replaced with MATH

3. The function MATH is continuous and nonexpansive: MATH

4. The distance function MATH is convex.

Proof

(1) follows from the theorem ( Weierstrass theorem ).

(2) We use notation MATH . Clearly, $x_{0}$ has to lie on the boundary of $C$ . Also, the $x_{0}$ has to satisfy the condition MATH where the $z$ is taken among all directions such that MATH remain in $C$ for small $\varepsilon>0$ . The differentiation reveals that MATH For any $y\in C$ the difference $y-x_{0}$ is a valid $z$ . Hence, the (2) follows.

(3) Since MATH we can write from (2) MATH We add the above and obtain MATH Hence, MATH

(4) follows from (3) and definition of convexity.





Notation. Index. Contents.


















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