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.
 P. Saddle point 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.

## Projection on convex set.

roposition

(Projection theorem). Let be a nonempty closed convex set.

1. For any there exists a unique vector called the projection of on .

2. The could be defined as the only vector with the property If the is affine and is a subspace parallel to then the above may be replaced with

3. The function is continuous and nonexpansive:

4. The distance function is convex.

Proof

(1) follows from the theorem ( Weierstrass theorem ).

(2) We use notation . Clearly, has to lie on the boundary of . Also, the has to satisfy the condition where the is taken among all directions such that remain in for small . The differentiation reveals that For any the difference is a valid . Hence, the (2) follows.

(3) Since we can write from (2) We add the above and obtain Hence,

(4) follows from (3) and definition of convexity.

 Notation. Index. Contents.