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.

## Recession cone.

efinition

A vector is a direction of recession of the set iff for , we have .

Directions of recession of a set constitute a cone that we denote . We introduce the notation The , if not empty, constitutes a subspace. We call it a "linearity space" of .

Proposition

(Main properties of direction of recession) Let be a closed convex set.

1. The vector is a direction of recession if contains for at least one .

2. is either compact or has a direction of recession.

To see that the closedness is necessary consider the set , see the figure ( Closedness and recession ). The only candidate for the direction of recession is . However, the point translates outside of along .

Closedness and recession

Proof

1. The statement (1) follows from the construction on the picture ( Direction of recession ). We start from the point and a direction of recession . We take any point and show that as follows.

For small enough sphere around if we take , then must be in . Then the limit is in . Hence, for small enough .

We conclude that for all by contradiction. If there is a finite then we step back for small enough and build an -sphere around as in the first part of this proof.

2. Take a point and assume existence of such that . A limit point of is a direction of recession.

Direction of recession.

Proposition

(Recession cone of intersection). Let and be closed convex sets and . Then .

To see that the requirement is necessary consider the sets and for , see the figure ( Closedness and recession ). These do not intersect but have a common direction of recession.

To see that the closedness is necessary consider and . The intersection is . It has a direction of recession . The has no direction of recession.

Proof

The statement ( Recession cone of intersection ) follows from ( Main properties of direction of recession -1) and the definitions.

Proposition

(Recession cone of inverse image). Let be a nonempty closed convex subset of , let be an matrix and let be a nonempty convex compact subset of . Assume that the set is nonempty. Then

Proof

By definition of we have . The sets and are convex and closed. Hence, the proposition ( Recession cone of intersection ) applies. It remains to note that .

Note that the compactness of is important. In absence of compactness we cannot state that the is closed and we cannot state that .

Proposition

(Decomposition of a convex set). For any subspace contained in for a non-empty convex set we have

Proof

Let be a subspace contained in . For any the affine set intersects . If then . Hence, the intersection is not empty. Then for some and .

 Notation. Index. Contents.