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

Directional derivative and subdifferential.


(Nondecreasing ratio). Let $I$ be an interval of $\QTR{cal}{R}$ and $f\left( x\right) $ is a convex function on $I$ . The function MATH is nondecreasing in each argument.


Observe that MATH . Hence, we assume $x\leq y$ without loss of generality. We aim to show that MATH for $x\leq y\leq z$ . There exists a $\theta$ such that MATH . We use such $\theta$ and the definition of convexity to calculate MATH


(Left and right derivatives). Let $f$ be a convex function on the interval MATH . The left and right derivatives $f^{-},f^{+}$ of $f$ are defined by MATH


(Properties of left and right derivative). Let $I$ be an interval MATH and let $f$ be a convex function on $I$ .

1. MATH .

2. If MATH then MATH and MATH are finite.

3. If MATH and $x\leq y$ then MATH .

4. The functions $f^{+},f^{-}$ are nondecreasing.


The statements are consequences of the proposition ( Nondecreasing ratio ).


(Directional derivative). For a function MATH the directional derivative is defined by MATH

Let $f$ be a convex function MATH . We use the notation MATH , MATH , $z\in\QTR{cal}{R}$ . Fix MATH . A hyperplane $H$ that passes through the point MATH and has the normal vector MATH is given by the relationship MATH Equivalently, MATH The MATH lies above $H$ iff MATH or

MATH (Subgradient)


(Subgradient and subdifferential). The vector MATH is a subgradient to the function $f$ at MATH iff the relationship ( Subgradient ) holds. The set of all subgradients at $x_{0}$ is called subdifferential at $x_{0}$ and denoted MATH .


(Existence of subdifferential). Let MATH be a convex function. For any MATH the MATH is nonempty, convex and compact set.


We match the conditions of the present proposition with the setup of the proposition ( Crossing theorem 2 ) as follows MATH Hence, according to the proposition ( Crossing theorem 2 ) MATH where MATH and $q^{\ast}$ is the maximal crossing point of the hyperplanes MATH such that the $M$ lies above the hyperplane MATH . Hence, there is a $\mu$ such that MATH or MATH Set $y=x_{0}+u$ then MATH Hence, $-\mu\,\ $ is a subgradient. The rest of the conclusions follow from the conclusions of the proposition ( Crossing theorem 2 ) and $D=\QTR{cal}{R}^{n}$ .

The following statements are verified with similar techniques.


(Properties of subgradient).

1. Let MATH be a convex function. For any MATH and any MATH we have MATH

2. For convex functions MATH MATH

3. For a $m\times n$ matrix $A$ MATH

4. Let $g$ be a smooth function MATH and MATH then MATH If $g$ is convex and nondecreasing then MATH


Let MATH be a proper convex function then MATH where the $S$ is a subspace parallel to MATH and $G$ is a nonempty compact set. Furthermore, MATH is nonempty and compact iff $x$ is in interior of MATH .


The proof of the proposition ( Existence of subdifferential ) applies almost without changes.

Notation. Index. Contents.

Copyright 2007