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.

## Polar cones.

efinition

(Polar cone definition). For a nonempty set we define the polar cone :

The following statement is a direct consequence of the definitions.

Proposition

(Polar cone properties). For any nonempty set , we have

1. is a closed convex set.

2. .

3. If for some set then .

Proposition

(Polar cone theorem). For any nonempty cone we have

If is closed and convex then .

Proof

First, we show that for any nonempty we have . Indeed, by the definitions, for a fixed Therefore, .

Next, we prove that for a closed nonempty , we have .

Let . Since is closed, there exists the projection . Let us translate the coordinate system so that . Then by the proposition ( Projection theorem )-2 we have Hence, We already established that Therefore, but also Hence, for a nonempty set (empty is a trivial case) we have By the definition of polar cone, we always have for a nonempty . Hence, .

Finally, we prove that . By the proposition ( Polar cone properties ), we have Therefore, We already proved that .

 Notation. Index. Contents.