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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
Author
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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.
a. Geometric multipliers.
b. Dual problem.
c. Connection of dual problem with minimax theory.
[. Conjugate duality.
VI. Basic Math II.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Geometric multipliers.


efinition

(Geometric multiplier). The pair MATH is a called a "geometric multiplier" for the problem ( Primal problem ) if $\mu^{\ast}\geq0$ and MATH

The following statement directly follows from the definitions ( Primal problem ),( Geometric multiplier

Proposition

(Visualization lemma). Assume MATH .

1. The hyperplane in MATH with normal MATH that passes through MATH also passes through MATH .

2. Among all hyperplanes with normal MATH that contain the set $S$ in the upper half-space, the highest level of intersection with the axis MATH is given by MATH .

Proposition

(Geometric multiplier property). Let MATH be a geometric multiplier then $x^{\ast}$ is a global minimum of the problem ( Primal problem ) if and only if $x^{\ast}\in C$ and MATH

Proof

Note that $x^{\ast}\in C$ implies MATH and MATH and the definition ( Geometric multiplier ) requires $\mu\leq0$ . Hence, Hence, MATH and MATH .

Let $x^{\ast}$ be a global minimum of the problem ( Primal problem ) then MATH By the definition ( Geometric multiplier ), MATH Therefore, MATH and MATH .

The statement is proven similarly in the other direction.

Definition

(Lagrange multiplier). The pair MATH is called "Lagrange multiplier of the problem ( Primal problem ) associated with the solution $x^{\ast}$ " if MATH and MATH

The following statement is a consequence of the proposition ( Local minimum of a sum ) and definitions.

Proposition

Assume that the problem ( Primal problem ) has at least one solution $x^{\ast}$ .

1. Let $f$ and MATH are either convex or smooth , MATH are smooth, $X$ is closed and MATH is convex then every geometric multiplier is a Lagrange multiplier.

2. Let $f$ and MATH are convex, MATH are affine and $X$ is closed and convex then the sets of Lagrange and geometric multiplier coincide.





Notation. Index. Contents.


















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