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.
VI. Basic Math II.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Recession cone.


efinition

A vector $y$ is a direction of recession of the set $C$ iff for $\forall x\in C$ , $\forall\lambda>0$ we have $x+\lambda y\in C$ .

Directions of recession of a set $C$ constitute a cone that we denote $R_{C}$ . We introduce the notation MATH The $L_{C}$ , if not empty, constitutes a subspace. We call it a "linearity space" of $C$ .

Proposition

(Main properties of direction of recession) Let $C$ be a closed convex set.

1. The vector $y$ is a direction of recession if $C$ contains MATH for at least one $x\in C$ .

2. $C$ is either compact or has a direction of recession.

To see that the closedness is necessary consider the set MATH , see the figure ( Closedness and recession ). The only candidate for the direction of recession is $\left( 0,1\right) $ . However, the point MATH translates outside of $C$ along $\left( 0,1\right) $ .


Closedness and recession figure
Closedness and recession

Proof

1. The statement (1) follows from the construction on the picture ( Direction of recession ). We start from the point $x$ and a direction of recession $y$ . We take any point $\bar{x}$ and show that MATH as follows.

For small enough sphere around $\bar{x}$ if we take MATH , MATH then MATH must be in $C$ . Then the limit $A=\lim_{n}a_{n}$ is in $C$ . Hence, MATH for small enough $\lambda>0$ .

We conclude that MATH for all $\lambda$ by contradiction. If there is a finite MATH then we step back MATH for small enough $\varepsilon$ and build an $2\varepsilon$ -sphere around $x_{0}$ as in the first part of this proof.

2. Take a point $y_{0}\in C$ and assume existence of $x_{n}\in C$ such that MATH . A limit point of MATH is a direction of recession.


Direction of recession figure
Direction of recession.

Proposition

(Recession cone of intersection). Let $X$ and $Y$ be closed convex sets and MATH . Then MATH .

To see that the requirement MATH is necessary consider the sets MATH and MATH for MATH , see the figure ( Closedness and recession ). These do not intersect but have a common direction of recession.

To see that the closedness is necessary consider $C_{0}$ and MATH . The intersection is MATH . It has a direction of recession $\left( 0,1\right) $ . The $C_{0}$ has no direction of recession.

Proof

The statement ( Recession cone of intersection ) follows from ( Main properties of direction of recession -1) and the definitions.

Proposition

(Recession cone of inverse image). $\ $ Let $C$ be a nonempty closed convex subset of $\QTR{cal}{R}^{n}$ , let $A$ be an $n\times m$ matrix and let $W$ be a nonempty convex compact subset of $\QTR{cal}{R}^{m}$ . Assume that the set MATH is nonempty. Then MATH

Proof

By definition of $V$ we have MATH . The sets MATH and $C$ are convex and closed. Hence, the proposition ( Recession cone of intersection ) applies. It remains to note that MATH .

Note that the compactness of $W$ is important. In absence of compactness we cannot state that the MATH is closed and we cannot state that MATH .

Proposition

(Decomposition of a convex set). For any subspace $S$ contained in $L_{C}$ for a non-empty convex set MATH we have MATH

Proof

Let $S$ be a subspace contained in $L_{C}$ . For any MATH the affine set $x+S$ intersects $S^{\perp}$ . If $x\in C$ then $x+S\subset C$ . Hence, the intersection MATH is not empty. Then $x=y+z$ for some $y\in S$ and MATH .





Notation. Index. Contents.


















Copyright 2007