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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.

Preservation of closeness under linear transformation.


he set MATH is a closed convex set. The projection on the $x$ -axis is a linear transformation. The image of $C$ under such transformation is open.

Proposition

(Preservation of closeness result). Let $C$ be a nonempty subset of $\QTR{cal}{R}^{n}$ and let $A$ be an $m\times n$ matrix.

1. If MATH then the set $AC$ is closed.

2. Let $X$ be a nonempty subset of $\QTR{cal}{R}^{n}$ given by linear constraints MATH If MATH then the set MATH is closed.

3. Let $C$ is given by the quadratic constraints MATH where the $Q_{j}$ are positive semidefinite matrices. Then the set $AC$ is closed.

Proof

(1). Let MATH , where the MATH is the ball around $z$ of radius $\varepsilon$ . The sets MATH are nested if MATH . It is suffice to prove that MATH is not empty for any sequence MATH

We have MATH Therefore, by the proposition ( Recession cone of inverse image ), MATH Consequently, in the context of the proposition ( Principal intersection result ) for MATH , MATH Since, generally MATH to accomplish the condition $R=L$ of the ( Principal intersection result ) it is enough to have MATH as required by the theorem.

Proof

(2). Let MATH . We introduce the sets MATH for MATH and aim to prove that the intersection MATH is not empty.

We have MATH By the propositions ( Recession cone of inverse image ) and ( Recession cone of intersection ) MATH Consequently, in the context of the proposition ( Principal intersection result ) for MATH , MATH Since, generally MATH to accomplish the condition $R=L$ of the ( Principal intersection result ) it is enough to have MATH

Proof

(3). Let MATH . We introduce the sets MATH for MATH and aim to prove that the intersection MATH is not empty.

We have MATH

We now apply the proposition ( Quadratic intersection result ) to conclude the proof.





Notation. Index. Contents.


















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