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.

## Polyhedral cones.

efinition

A cone is polyhedral if it has the form

A cone is finitely generated if it has the form where .

Proposition

(Polar polyhedral cone). Let . Then is closed and

Proof

First we prove that . Indeed, by the definition of polar cone

Next, we prove that is close by induction in .

For it is closed.

We assume that is closed and prove that is closed. Without loss of generality we assume .

Take any sequence . We aim to prove that . We have The must be a bounded sequence. Hence, we take a subsequence converging to some limit point and restrict consideration to such subsequence: We have Therefore, must be convergent: and by the induction hypothesis. Hence,

Proposition

(Farkas lemma). Let where .

Then

Proof

Note that Therefore, by the proposition ( Polar polyhedral cone ), and is closed. Hence,

Proposition

(Minkowski-Weyl theorem). A cone is polyhedral if and only if it is finitely generated.

Proof

Suppose is a finitely generated cone We prove that there exist vectors such that

Let be a linear span of , and . We introduce to be the orthogonal basis of . Hence we have defined the linear transformations and as follows The transformation is known as "orthogonalization". Some of its columns have all zero elements because might be linearly dependent.

We have Let We introduce the vectors : then Therefore,

Definition

A set is a polyhedral set if it is nonempty and has the form

Proposition

(Minkowski-Weyl representation). A set is polyhedral iff for some .

Proof

Note that the inequality may be represented as Based on this observation we aim to apply the proposition ( Minkowski-Weyl theorem ). Set of the form is not cone. We consider Observe that . By the proposition ( Minkowski-Weyl theorem ), we have We introduce the notation We have

Definition

A function is polyhedral if is polyhedral.

The following proposition is a direct consequence of the definition.

Proposition

(Polyhedral function). Let be a convex function. Then is polyhedral if and only if is polyhedral and

 Notation. Index. Contents.