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.

Lagrange multipliers for equality constraints.

e are considering the following problem.

 (Minimization with equality constraints)
where the and are smooth functions .

Proposition

(Existence of Lagrange multipliers for equality constraints). Let be a local minimum of the problem ( Minimization with equality constraints ) and then there are scalars such that

Proof

The condition implies where the matrix consists of the columns Hence, Equivalently, Therefore, We next show that Indeed, we already established that is a subspace, hence, . Therefore, if then . But and we proved already that

Therefore, According to the proposition ( Minimum of a smooth function ), hence, and the conclusion of the proposition follows.

The condition states that the consists of directions tangent to the level surfaces of crossing the . For example,

 Notation. Index. Contents.