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.

## Directional derivative and subdifferential.

roposition

(Nondecreasing ratio). Let be an interval of and is a convex function on . The function is nondecreasing in each argument.

Proof

Observe that . Hence, we assume without loss of generality. We aim to show that for . There exists a such that . We use such and the definition of convexity to calculate

Definition

(Left and right derivatives). Let be a convex function on the interval . The left and right derivatives of are defined by

Proposition

(Properties of left and right derivative). Let be an interval and let be a convex function on .

1. .

2. If then and are finite.

3. If and then .

4. The functions are nondecreasing.

Proof

The statements are consequences of the proposition ( Nondecreasing ratio ).

Definition

(Directional derivative). For a function the directional derivative is defined by

Let be a convex function . We use the notation , , . Fix . A hyperplane that passes through the point and has the normal vector is given by the relationship Equivalently, The lies above iff or

Definition

(Subgradient and subdifferential). The vector is a subgradient to the function at iff the relationship ( Subgradient ) holds. The set of all subgradients at is called subdifferential at and denoted .

Proposition

(Existence of subdifferential). Let be a convex function. For any the is nonempty, convex and compact set.

Proof

We match the conditions of the present proposition with the setup of the proposition ( Crossing theorem 2 ) as follows Hence, according to the proposition ( Crossing theorem 2 ) where and is the maximal crossing point of the hyperplanes such that the lies above the hyperplane . Hence, there is a such that or Set then Hence, is a subgradient. The rest of the conclusions follow from the conclusions of the proposition ( Crossing theorem 2 ) and .

The following statements are verified with similar techniques.

Proposition

1. Let be a convex function. For any and any we have

2. For convex functions

3. For a matrix

4. Let be a smooth function and then If is convex and nondecreasing then

Proposition

Let be a proper convex function then where the is a subspace parallel to and is a nonempty compact set. Furthemore, is nonmepty and compact iff is in interior of .

Proof

The proof of the proposition ( Existence of subdifferential ) applies almost without changes.

 Notation. Index. Contents.