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.
 a. Geometric multipliers.
 b. Dual problem.
 c. Connection of dual problem with minimax theory.
 [. Conjugate duality.
 VI. Basic Math II.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Geometric multipliers.

efinition

(Geometric multiplier). The pair is a called a "geometric multiplier" for the problem ( Primal problem ) if and

The following statement directly follows from the definitions ( Primal problem ),( Geometric multiplier

Proposition

(Visualization lemma). Assume .

1. The hyperplane in with normal that passes through also passes through .

2. Among all hyperplanes with normal that contain the set in the upper half-space, the highest level of intersection with the axis is given by .

Proposition

(Geometric multiplier property). Let be a geometric multiplier then is a global minimum of the problem ( Primal problem ) if and only if and

Proof

Note that implies and and the definition ( Geometric multiplier ) requires . Hence, Hence, and .

Let be a global minimum of the problem ( Primal problem ) then By the definition ( Geometric multiplier ), Therefore, and .

The statement is proven similarly in the other direction.

Definition

(Lagrange multiplier). The pair is called "Lagrange multiplier of the problem ( Primal problem ) associated with the solution " if and

The following statement is a consequence of the proposition ( Local minimum of a sum ) and definitions.

Proposition

Assume that the problem ( Primal problem ) has at least one solution .

1. Let and are either convex or smooth , are smooth, is closed and is convex then every geometric multiplier is a Lagrange multiplier.

2. Let and are convex, are affine and is closed and convex then the sets of Lagrange and geometric multiplier coincide.

 Notation. Index. Contents.