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.

Fritz John optimality conditions.

roblem

(Smooth optimization problem). We consider the following problem where the are smooth functions and is a nonempty closed set.

Proposition

(Fritz John conditions). Let be a local minimum of the problem ( Smooth optimization problem ). Then there exist quantities such that

1.

2.

3.

4. Let If then there exists such that where and

Proof

Let for

Consider the problems

 (Penalized problem)
where and is such that

By the classic version of the Weierstrass theorem there exists a solution of the problem ( Penalized problem ) for every . In particular, Note that and . Hence, we rewrite the last inequality as

 (FJ proof 1)
By construction, is a bounded sequence. Therefore, is has one or more limit points .

The is smooth, hence, is bounded. Therefore, because otherwise and cannot be bounded by the .

Thus, all the limit points are feasible: Therefore, by the construction of and ,

 (FJ proof 2)
By passing to the limit the inequality ( FJ proof 1 ) and combining with ( FJ proof 2 )we conclude for every limit point . Thus is convergent and

According to the proposition ( Minimum of a smooth function ) By convergence , for large enough the is inside , hence We restrict our attention to such .

We calculate

 (FJ proof 3)
and introduce the notation Note that the sequence of is bounded: Hence, it has a limit point

By dividing ( FJ proof 3 ) with we obtain We pass the last relationship to the limit and arrive to

(compare with the definition ( Normal cone )).

To see that the statement (4) holds, note that by construction of , , if then for large enough . If then the -th component of : has to vanish as . Hence, if then vanishes quicker than any of the for . The consideration for is identical.

 Notation. Index. Contents.