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.
 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.

## Dual problem. roblem

(Dual problem). Find where The dual problem delivers the highest crossing point for the set Note that is an of a collection of affine functions. Hence, it is concave, upper semi-continuous and may be studied with the means of the propositions ( Crossing theorem 1 ),( Crossing theorem 2 ). In particular, the following statement directly follows from the proposition ( Crossing theorem 1 ), the geometrical interpretation of the ( Visualization lemma ) and the definition ( Geometric multiplier ).

Proposition

(Duality gap and geometric multipliers). The following alternative takes place.

1. If (="there is no duality gap") then the set of geometric multipliers is equal to the set of solutions of the problem ( Dual problem ).

2. If (="there is a duality gap") then the set of geometric multipliers is empty.

 Notation. Index. Contents.