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.

## Weierstrass Theorem.

continuous function attains its minimum on a compact set. Such statement is the simplest version of the Weierstrass theorem. In this section we prove an extended version. We need some preliminary results and definitions.

Definition

(Limit points) Let be a sequence of real numbers.

1. Let . We introduce the notation

If is not bounded from above then we write .

If is not bounded from below then we write .

2. The point is a limit point of the sequence is there is an infinite number of points from in an -neighborhood of for any .

Proposition

The is the greatest limit point of the sequence . The is the smallest limit point of the sequence .

Definition

A function is proper if its epigraph is nonempty and does not contain a vertical line. The function is closed if the is a closed set.

Definition

A function is lower semicontinuous if for any and we have

Proposition

(Closeness and lower semicontinuity). Let be a function . The following statements are equivalent:

1. The level sets are closed for every .

2. The function is lower semicontinuous.

3. The is a closed set.

Proof

(1) implies (2). Since the level sets are closed we have that for any sequence and vector such that and we must also have . Assume that (2) is not true. Then there exists a and such that and for some scalar . This constitutes a contradiction with the noted consequence of closeness of the level sets.

The rest may be proved with similar means.

Proposition

(Weierstrass theorem). Let be a closed proper function. If any of the below three conditions holds then the set is nonempty and compact.

1. is bounded.

2. There exists an such that the level set is nonempty and bounded.

3. If then .

Proof

1. Let be a sequence such that . Since is bounded the sequence has a limit point . By proposition ( Closeness and lower semicontinuity ) the is lower semicontinuous. Hence, . Therefore, is nonempty. The is an intersection of level sets. Hence, the compactness of follows from the boundedness of and closeness of the level sets.

The (2) proves similarly to (1).

The (3) implies (2).

 Notation. Index. Contents.