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.

## Feasible direction cone, tangent cone and normal cone.

efinition

Let be a subset of and be a point in .

(Feasible direction cone). The feasible direction cone of at is defined as follows.

(Tangent cone). The tangent cone of at is defined as follows

(Normal cone). The normal cone of at is defined as follows

(Regularity of a set). By definition, the is regular at if

Tangent cone figure 1

On the figure ( Tangent cone figure 1 ) the is the closed area bounded by the circle, the is the origin, the and .

Tangent cone figure 2

On the figure ( Tangent cone figure 2 ) the is the curved line, the is the origin, the and .

Normal cone figure 1

On the figure ( Normal cone figure 1 ) the is the closed area bounded by the curved shape, the is the origin, , and . To see that note that the condition of the definition ( Normal cone ) requires that approach along the boundary of . For any other choice of we have and .

Proposition

(Tangent cone 2). Let be a subset of and . Then

Proof

Let then according to the definition ( Tangent cone ) there is a sequence s.t. and . We set .

Conversely, let be the sequence as stated in the proposition then and

Proposition

(Tangent cone 3). Let be a subset of and .

1. is a closed cone.

2. .

3. If is convex then and are convex and

Proof

(1). Consider such that . We aim to show that . We exclude non essential case .

By definition of there are sequences , and as .

There exists an increasing function s.t. . We can also find a function such that , and . The sequence is the sequence that we need to show that in context of the definition ( Tangent cone ).

Proof

(2). by definitions and by (1) the is closed.

Proof

(3). Since is convex all the feasible directions are of the form , . Hence, is convex. By the proposition ( Tangent cone 2 ) the consists of that are limit points sequences of such feasible directions . Hence, . Therefore, in combination with (2), the follows and is convex.

Proposition

(Tangent cone 4). Let be a nonempty convex subset of and .

1. .

2. is regular for all : .

3. .

Proof

Since is convex, any feasible direction is of the form . Hence, (1) follows from the proposition ( Tangent cone 3 )-3 and the definition ( Polar cone definition ).

The (2) follows from (1) and the definition ( Normal cone ).

The (3) is a consequence of the proposition ( Polar cone theorem ), (2) and the proposition ( Tangent cone 3 )-1,3.

 Notation. Index. Contents.