Quantitative Analysis
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Numerical Analysis
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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.
P. Saddle point 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.

Existence of solution of convex optimization problem.


roposition

(Directions of recession). Let MATH be a closed proper convex function.

1. All nonempty level sets MATH have the same recession cone given by MATH

2. If one nonempty level set is compact then all the level sets are compact.

Proof

Given a direction $y$ and a point $x$ the function MATH is either nonincreasing or increasing starting from some large enough $\lambda$ . If it is nonincreasing then $y$ is in MATH for any $\gamma$ . The rest follows from the proposition ( Main properties of direction of recession ).

Proposition

(Basic existence result). Let $X$ be a closed convex subset of $R^{n}$ and let MATH be a closed proper convex function such that MATH . The set MATH is nonempty and compact if and only if the $X$ and $f$ have no common directions of recession.

If $X$ and $f$ has no common direction of recession then the minimum cannot escape to infinity. Such intuition may be formalized into a proof by considering intersections of the nested compact convex sets MATH with the sequence MATH converging to the MATH . The following proposition is a consequence of the same observation and the propositions ( Principal intersection result ),( Linear intersection result ) and ( Quadratic intersection result ).

Proposition

(Unbounded existence result). Let $X$ be a closed convex subset of $\QTR{cal}{R}^{n}$ and let MATH be a closed proper convex function such that MATH . The set MATH is nonempty if any of the following conditions hold.

1. MATH .

2. MATH and $X$ is given by the linear constraints MATH for some $a_{j},b_{j}$ .

3. MATH and $f,X$ are of the form MATH where the $Q,Q_{j}$ are positive semidefinite matrixes.

Remark

The convex function $f$ is constant on the subspace $L_{f}$ .





Notation. Index. Contents.


















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