Content of present website is being moved to www.lukoe.com/finance . Registration of www.opentradingsystem.com will be discontinued on 2020-08-14.
 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.
 a. Affine sets and hyperplanes.
 b. Convex sets and cones.
 c. Convex functions and epigraphs.
 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.
 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.

## Convex functions and epigraphs.

efinition

(Convex and proper function). The "epigraph" of a function is the set , see the picture ( Picture of convex function ). The function is "convex" iff the set is convex. The "effective domain" is the set .

The function is "proper" if the epigraph is nonempty and does not contain a vertical line.

Convex function acting from to . Level sets .

The consideration of this entire chapter on convex analysis is restricted to proper functions. Hence, all functions that are said to be convex are also presumed to be proper.

Proposition

(Main property of convex function). A function is convex iff .

Proposition

A smooth function is convex iff the matrix of second derivatives is non-negatively determined.

Proof

Fix two points and and denote Let be the matrix of second derivatives taken at the point . Note that We use the Taylor decomposition If we assume that the function is convex then we have, by the proposition ( Main property of convex function ), Hence, for . Therefore, for any .

Proposition

(Preservation of convexity).

1. If are convex functions and are positive real numbers then is convex.

2. If is a convex function and is a matrix then is convex.

3. If are convex functions and is an arbitrary index set then is convex.

 Notation. Index. Contents.