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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.
 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.

Affine sets and hyperplanes.

efinition

The set is called "the line through and ". The set is called "affine" iff for . The operation on set is called "translation of by ". If is a subspace then the set is called "parallel to ".

Subspace is an affine set containing the origin. Every affine set is a translation of some subspace.

Definition

Dimension of the affine set is the dimension of the parallel subspace. Affine sets of dimension are called "hyperplanes". The set is called "orthogonal complement of ."

Proposition

(Hyperplane representation)Hyperplanes are sets of the form .

Proof

Subspaces of dimension are orthogonal complements of vectors. Hyperplanes are translations of such subspaces.

Proposition

Affine sets have the form where is a matrix and is a vector. Consequently, affine sets are intersections of hyperplanes.

Proof

If is an affine set then for some subspace . Let be the basis of then . We set as a union of columns and .

Intersection of affine sets is an affine set. Hence, we introduce the affine hull as follows.

Definition

(Affine hull). The affine hull of the set is .

 Notation. Index. Contents.