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.

## Preservation of closeness under linear transformation.

he set is a closed convex set. The projection on the -axis is a linear transformation. The image of under such transformation is open.

Proposition

(Preservation of closeness result). Let be a nonempty subset of and let be an matrix.

1. If then the set is closed.

2. Let be a nonempty subset of given by linear constraints If then the set is closed.

3. Let is given by the quadratic constraints where the are positive semidefinite matrices. Then the set is closed.

Proof

(1). Let , where the is the ball around of radius . The sets are nested if . It is suffice to prove that is not empty for any sequence

We have Therefore, by the proposition ( Recession cone of inverse image ), Consequently, in the context of the proposition ( Principal intersection result ) for , Since, generally to accomplish the condition of the ( Principal intersection result ) it is enough to have as required by the theorem.

Proof

(2). Let . We introduce the sets for and aim to prove that the intersection is not empty.

We have By the propositions ( Recession cone of inverse image ) and ( Recession cone of intersection ) Consequently, in the context of the proposition ( Principal intersection result ) for , Since, generally to accomplish the condition of the ( Principal intersection result ) it is enough to have

Proof

(3). Let . We introduce the sets for and aim to prove that the intersection is not empty.

We have

We now apply the proposition ( Quadratic intersection result ) to conclude the proof.

 Notation. Index. Contents.