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.

Minimal common and maximal crossing points.

et be a subset of . The may have common points with the -th coordinate axis. We introduce the quantity

A normal vector to a nonvertical hyperplane may be normalized to a form . A nonvertical hyperplane that crosses the -th coordinate axis at the point and has a normal vector has the representation Indeed,

The set is contained in the upper half plane of iff Hence, the quantity is the maximum -th axis crossing level for all hyperplanes that contain the set in the upper half space and have the normal vector .

The is a concave function.

We introduce the quantity

Proposition

(Weak duality theorem). Let be a subset of . Then

Proof

.

We investigate the conditions for the equality Observe that by definition of these quantities all that is needed is existence of a supporting hyperplane at a point . The pictures ( Crossing points figure 1 )-( Crossing points figure 3 ) show basic examples when this may or may not happen.

Crossing points figure 1

Crossing points figure 2. The upper boundary is included in the set. The other boundaries are excluded.

Crossing points figure 3

Proposition

(Crossing theorem 1). Let be a subset of . Assume the following:

1. and -th axis have nonempty intersection and .

2. The set is convex.

Then if and only if for any sequence such that we have

Proof

By definition of , .

contains no vertical lines. Indeed, if it does then by the proposition ( Main properties of direction of recession ) one may infinitely go along the vector inside starting from any . This contradicts the condition 1.

We have for any small positive . Indeed, on the contrary, if then by definition of the closure one can construct the sequence that violates .

Therefore, by the proposition ( Nonvertical separation ), there is a nonvertical separation of from for any small positive . The -th axis crossing point for such separating hyperplane must be between and . Hence, .

Proposition

(Crossing theorem 2). Let be a subset of . Assume the following:

1. and -th axis have nonempty intersection and .

2. The set is convex.

3. , where the set is defined by Then and the solution set has the form where the set is nonempty convex and compact and is the orthogonal complement of relative to the plane of the first n coordinates .

Crossing theorem 2 figure

Proof

By the proposition ( Proper separation 1 ) there is a separating hyperplane for the point and set . Such hyperplane cannot be vertical. Indeed, if it is vertical then the point projects on the plane along the onto the origin . Indeed, the segment would belong to . But then the condition is violated because it would belong to the boundary of . Therefore the is nonvertical, and is nonempty.

We next claim that . Indeed, by construction of if it has an orthogonal complement in then we can rotate coordinate system to make a coordinate subspace and then remove the coordinates that span the from the consideration (see the picture ( Crossing theorem 2 figure )).

In addition, . To see this, consider any hyperplane , corresponding normal that delivers and the perturbation . If then can be made arbitrarily close to horizontal and would be close to vertical by taking large enough . Hence, such can be in only if . If then the statement is trivially true. We exclude such case from consideration.

We conclude that .

We next apply the proposition ( Decomposition of a convex set ) within the with . The and have no common direction of recession as we already established. Hence, for some convex and nonempty . The is compact by ( Main properties of direction of recession )-2.

 Notation. Index. Contents.