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Quantitative Analysis
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Numerical Analysis
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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.

Saddle point theory.


et $X$ and $Z$ be nonempty convex subsets of $\QTR{cal}{R}^{n}$ and $\QTR{cal}{R}^{m}$ respectively and let $\phi$ be a function MATH . We introduce the following notations. MATH MATH MATH MATH

The following statements are consequences of the propositions ( Minimax theorem ) and ( Partial minimization result ).

Proposition

(Saddle point result 1) Assume that

1. $\forall z\in Z$ the function $t_{z}$ is convex and closed,

2. $\forall x\in X$ the function $r_{x}$ is convex and closed,

3. MATH .

Then the minimax equality MATH holds and MATH is nonempty under any of the following conditions.

0. The level sets of the function $t$ are compact.

1. The recession cone and the constancy space of the function $t$ are equal.

2. The function MATH has the form MATH with $\bar{F}$ being a closed proper convex function and set $C$ being given by the linear constraints MATH

and MATH .

3. MATH MATH where $Q,R$ are symmetric matrices, $Q$ is positive semidefinite, $R$ is positive definite, MATH where the $Q_{j}$ are positive semidefinite matrixes.

In addition, if (0) holds then $X^{\ast}$ is compact.

Proposition

(Saddle point result 2). Assume that

1. $\forall z\in Z$ the function $t_{z}$ is convex and closed,

2. $\forall x\in X$ the function $r_{x}$ is convex and closed,

3. Either MATH or MATH .

Then

(a) If the level sets of functions $r$ and $t$ are compact then the set of saddle points of $\phi$ is nonempty and compact.

(b) If $R_{r}=L_{r}$ and $R_{t}=L_{t}$ then the set of saddle points of $\phi$ is nonempty.

Proposition

(Saddle point theorem). Assume that

1. $\forall z\in Z$ the function $t_{z}$ is convex and closed,

2. $\forall x\in X$ the function $r_{x}$ is convex and closed,

Then the set of saddle points of $\phi$ is nonempty and compact if any of the following conditions are satisfied

1. $X$ and $Y$ are compact.

2. $Z$ is compact and MATH is nonempty and compact for some $\bar{z}\in Z$ and $\gamma$ .

3. $X$ is compact and MATH is nonempty and compact for some $\bar{x}\in X$ and $\gamma$ .

4. MATH and MATH are nonempty and compact for some $\bar{z}\in Z$ , $\bar{x}\in X$ and $\gamma$ .





Notation. Index. Contents.


















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