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.
 a. Support function.
 b. Infimal convolution.
 VI. Basic Math II.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Conjugate duality.

roposition

For any convex function

Proof

Each affine function corresponds to a hyperplane. By the proposition ( Supporting hyperplane theorem ), for any point below epigraph of there exists a hyperplane that separates such point from the . Hence, for any such point there exists a pair s.t. and for .

is the intersection of the upper half planes.

Corollary

The set is equal to intersection of the upper half-planes defined by the hyperplanes from proof of the previous statement.

Let us denote . Let us introduce a set Such set is not empty and it is an epigraph of some function because if then for all . Let us denote such function . By definition

Geometrical meaning of .

Let us introduce Observe that . Indeed, means that or where the and run through all the hyperplanes that define . For the same reason the infimum of such is the :

Summary

(Conjugate duality theorem). We define the operation of taking a dual function by Then for all proper convex functions.

The closure part: comes from the fact that taking affine envelopes includes boundary points of the into the final result .

 a. Support function.
 b. Infimal convolution.
 Notation. Index. Contents.
 Copyright 2007