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 sets and cones.

efinition

(Convex set). The set is called "the line segment between and ". The set is called "convex" iff for . Dimension of the convex set is the dimension of its affine hull.

Intersection of convex sets is a convex set. Consequently, for any collections of numbers and points the set is convex.

Definition

(Convex hull). The convex hull of any set is the intersection of all convex sets that contain . If the collection of numbers is such that and then the sum is called "the convex combination of points ".

Proposition

The convex hull of set S consists of all convex combinations of all elements of S.

Definition

(Convex Hull Cone Relative Interior). The set is called a "cone" if it is closed with respect to positive scalar multiplication: for and . The convex cone "generated by the set and denoted " " is the convex hull of all the lines joining all points of with the origin. Let denote a unit ball. The "closure" of the set is the set . The "relative interior" is the set .

 Notation. Index. Contents.