Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
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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.

Partial minimization of convex functions.


(Convexity of partial minimum). Let MATH be a convex function. Then the function $f$ given by MATH is convex.

The proof of the above proposition is a direct verification based on definitions.

The study of closeness of the partial minimum is based on the following observation.

Suppose the level set MATH is nonempty for some $\gamma$ . Let MATH be a sequence such that MATH . Then MATH The set MATH is closed if MATH is closed. The set MATH is a projection of MATH . Its closeness may be studied by means of the proposition ( Preservation of closeness result ). The intersection preserves the closeness. Hence, we arrive to the following proposition.


(Partial minimization result). Let MATH be a closed proper convex function. Then the function MATH is closed, convex and proper if any of the following conditions hold.

0. There exist MATH and MATH such that MATH is nonempty and compact.

1. There exist MATH and MATH such that MATH is nonempty and MATH .

2. MATH ,for some $\bar{F}$ , where the $C$ is given by the linearity constraints MATH and there exists $\bar{x}$ such that MATH

3. MATH , where the $C$ is given by the quadratic constraints MATH where the $Q_{j}$ are positive semidefinite and there exists $\bar{x}$ such that MATH

Notation. Index. Contents.

Copyright 2007