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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.
[. Conjugate duality.
VI. Basic Math II.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Lagrange multipliers for equality constraints.


e are considering the following problem.

MATH (Minimization with equality constraints)
where the $f$ and $h_{i}$ are smooth functions MATH .

Proposition

(Existence of Lagrange multipliers for equality constraints). Let $x^{\ast}$ be a local minimum of the problem ( Minimization with equality constraints ) and MATH then there are scalars MATH such that MATH

Proof

The condition MATH implies MATH where the $n\times m$ matrix $A$ consists of the columns MATH MATH Hence, MATH Equivalently, MATH Therefore, MATH We next show that MATH Indeed, we already established that MATH is a subspace, hence, MATH . Therefore, if MATH then MATH . But MATH and we proved already that MATH

Therefore, MATH According to the proposition ( Minimum of a smooth function ), MATH hence, MATH and the conclusion of the proposition follows.

The condition MATH states that the MATH consists of directions tangent to the level surfaces of $h_{i}$ crossing the $x^{\ast}$ . For example, MATH





Notation. Index. Contents.


















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