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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.

Feasible direction cone, tangent cone and normal cone.


efinition

Let $X$ be a subset of $\QTR{cal}{R}^{n}$ and $x$ be a point in $X$ .

(Feasible direction cone). The feasible direction cone MATH of $X$ at $x$ is defined as follows. MATH

(Tangent cone). The tangent cone MATH of $X$ at $x$ is defined as follows MATH

(Normal cone). The normal cone MATH of $X$ at $x$ is defined as follows MATH

(Regularity of a set). By definition, the $X$ is regular at $x$ if MATH


Tangent cone figure 1
Tangent cone figure 1

On the figure ( Tangent cone figure 1 ) the $X$ is the closed area bounded by the circle, the $x$ is the origin, the MATH and MATH .


Tangent cone figure 2
Tangent cone figure 2

On the figure ( Tangent cone figure 2 ) the $X$ is the curved line, the $x$ is the origin, the MATH and MATH .


Normal cone figure 1
Normal cone figure 1

On the figure ( Normal cone figure 1 ) the $X$ is the closed area bounded by the curved shape, the $x$ is the origin, MATH , MATH and MATH . To see that MATH note that the condition MATH of the definition ( Normal cone ) requires that MATH approach MATH along the boundary of $X$ . For any other choice of MATH we have MATH and MATH .

Proposition

(Tangent cone 2). Let $X$ be a subset of $\QTR{cal}{R}^{n}$ and $x\in X$ . Then MATH

Proof

Let MATH then according to the definition ( Tangent cone ) there is a sequence MATH s.t. $x_{k}\rightarrow x$ and MATH . We set MATH .

Conversely, let MATH be the sequence as stated in the proposition then MATH and MATH

Proposition

(Tangent cone 3). Let $X$ be a subset of $\QTR{cal}{R}^{n}$ and $x\in X$ .

1. MATH is a closed cone.

2. MATH .

3. If $X$ is convex then MATH and MATH are convex and MATH

Proof

(1). Consider MATH such that $y_{k}\rightarrow y$ . We aim to show that MATH . We exclude non essential case $y=0$ .

By definition of MATH there are sequences MATH , MATH and MATH as $p\rightarrow\infty$ .

There exists an increasing function MATH s.t. MATH . We can also find a function MATH such that MATH , MATH and MATH . The sequence MATH is the sequence that we need to show that MATH in context of the definition ( Tangent cone ).

Proof

(2). MATH by definitions and by (1) the MATH is closed.

Proof

(3). Since $X$ is convex all the feasible directions MATH are of the form MATH , $\alpha>0$ . Hence, MATH is convex. By the proposition ( Tangent cone 2 ) the MATH consists of $y$ that are limit points sequences of such feasible directions MATH . Hence, MATH . Therefore, in combination with (2), the MATH follows and MATH is convex.

Proposition

(Tangent cone 4). Let $X$ be a nonempty convex subset of $\QTR{cal}{R}^{n}$ and $x\in X$ .

1. MATH .

2. $X$ is regular for all $x\in X$ : MATH .

3. MATH .

Proof

Since $X$ is convex, any feasible direction MATH is of the form MATH . Hence, (1) follows from the proposition ( Tangent cone 3 )-3 and the definition ( Polar cone definition ).

The (2) follows from (1) and the definition ( Normal cone ).

The (3) is a consequence of the proposition ( Polar cone theorem ), (2) and the proposition ( Tangent cone 3 )-1,3.





Notation. Index. Contents.


















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