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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.

Weierstrass Theorem.


continuous function attains its minimum on a compact set. Such statement is the simplest version of the Weierstrass theorem. In this section we prove an extended version. We need some preliminary results and definitions.

Definition

(Limit points) Let MATH be a sequence of real numbers.

1. Let MATH . We introduce the notation MATH

If MATH is not bounded from above then we write MATH .

If MATH is not bounded from below then we write MATH .

2. The point $x_{0}$ is a limit point of the sequence MATH is there is an infinite number of points from MATH in an $\varepsilon$ -neighborhood of $x_{0}$ for any $\varepsilon>0$ .

Proposition

The MATH is the greatest limit point of the sequence MATH . The MATH is the smallest limit point of the sequence MATH .

Definition

A function MATH is proper if its epigraph is nonempty and does not contain a vertical line. The function $f$ is closed if the MATH is a closed set.

Definition

A function $f$ is lower semicontinuous if for any $x$ and MATH we have MATH

Proposition

(Closeness and lower semicontinuity). Let $f$ be a function MATH . The following statements are equivalent:

1. The level sets MATH are closed for every MATH .

2. The function $f$ is lower semicontinuous.

3. The MATH is a closed set.

Proof

(1) implies (2). Since the level sets are closed we have that for any sequence MATH and vector $x$ such that $x_{n}\rightarrow x$ and MATH we must also have MATH . Assume that (2) is not true. Then there exists a $y$ and MATH such that $y_{k}\rightarrow y$ and MATH for some scalar $\alpha$ . This constitutes a contradiction with the noted consequence of closeness of the level sets.

The rest may be proved with similar means.

Proposition

(Weierstrass theorem). Let MATH be a closed proper function. If any of the below three conditions holds then the set MATH is nonempty and compact.

1. MATH is bounded.

2. There exists an MATH such that the level set MATH is nonempty and bounded.

3. If MATH then MATH .

Proof

1. Let MATH be a sequence such that MATH . Since MATH is bounded the sequence $x_{k}$ has a limit point $x^{\ast}$ . By proposition ( Closeness and lower semicontinuity ) the $f$ is lower semicontinuous. Hence, MATH . Therefore, $\arg\min f$ is nonempty. The $\arg\min f$ is an intersection of level sets. Hence, the compactness of $\arg\min f$ follows from the boundedness of MATH and closeness of the level sets.

The (2) proves similarly to (1).

The (3) implies (2).





Notation. Index. Contents.


















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