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.
 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.

## Hyperplanes and separation.

efinition

Hyperplane in is a set of the form The is called the "normal vector". The sets are called "closed half-spaces" associated with .

The two sets and are separated by if either or

The two sets and are strictly separated by if the above inequalities are strict.

A hyperplane may be represented as for any fixed .

Proposition

(Supporting hyperplane theorem). Let be a nonempty convex subset of and . If does not belong to interior of then there is a hyperplane that passes through and contains in one of its closed half-spaces:

Proof

If then we obtain the normal vector by projecting on The may be obtained from the requirement that the pass trough .

If by does not belong to interior of then there is a sequence such that and . We utilize the construction from the case to obtain a sequence . The may be normalized to unity. The then has a limit point. Such limit point delivers the sought out hyperplane because of the proposition ( Projection theorem )-2.

Proposition

(Separating hyperplane theorem). If the are two nonempty disjoint convex sets then there is a hyperplane that separates them.

Proof

Apply the proposition ( Supporting hyperplane theorem ) to the set and .

Two nonempty convex disjoint sets are not necessarily strictly separated. For example, , do not have a strictly separating hyperplane.

Proposition

(Strict hyperplane separation 1). Let and are two nonempty convex disjoint sets. If is closed then there is a strictly separating hyperplane.

Proof

Let , , . Set By the closedness, . The strictly separates .

Let and be two disjoint closed convex subsets of . To investigate the conditions for to be closed we introduce the subset of , note that the transformation is linear and seek to apply the proposition ( Preservation of closeness result ). We note that is closed and convex, and The condition of the proposition ( Preservation of closeness result ) becomes

We arrive to the following additional sufficient conditions for strict separation.

Proposition

(Strict hyperplane separation 2). Let and are two nonempty convex disjoint sets. There is a strictly separating hyperplane if any of the following conditions holds.

1. is closed and is compact.

2. are closed and

3. is closed, is given by the linearity constraints and .

4. and are given by quadratic constraints where the are positive semidefinite matrices.

Proposition

(Intersection of half-spaces). The closure of convex hull of a set is the intersection of all closed half-spaces that contain .

Proof

If there is a point in that is not contained in the intersection of half-planes then we arrive to contradiction by using the theorem ( Supporting hyperplane theorem ).

Definition

The subsets of are properly separated by a hyperplane if the following conditions are true

Let be a line for some . The definition of the proper separation requires that is a single point or nothing and consists of more then one point.

The sets and may not be properly separated.

The sets and are properly separated by the x-axis .

Proposition

(Proper separation 1) Let is a subset of and . There is a properly separating hyperplane for and if

Proof

If then and are strictly separated by the proposition ( Strict hyperplane separation 2 )-2.

If then we translate into a subspace and apply the ( Strict hyperplane separation 2 )-2 within to obtain some separating plane then extend it to a hyperplane by .

Proposition

(Proper separation 2) The two subsets of are properly separated if

Proof

Apply the proposition ( Proper separation 1 ) to and .

 Notation. Index. Contents.