Content of present website is being moved to www.lukoe.com/finance . Registration of www.opentradingsystem.com will be discontinued on 2020-08-14.
 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.

## Optimality conditions.

roposition

(Minimum of a smooth function). Let be a smooth function and let be a minimum of over the subset of . Then Equivalently, If is convex then If then

Proof

Let , then there exists such that By smoothness of we have Hence, We pass the above to the limit and obtain The rest of the proposition follows from the proposition ( Tangent cone 4 )-1.

Proposition

(Minimum of a convex function). Let be a convex function and let be a convex subset of . Then

Equivalently,

Proof

Assume and for any . Then by the definition ( Subgradient and subdifferential ) and thus .

Conversely, let . Then for any . According to the proposition ( Properties of subgradient )-1 According to the proposition ( Existence of subdifferential ) the is taken over a compact set. Also, the is a continuous function of . Hence, the is achieved at some . Such has the property

The second part of the proposition is evident because the statement may be rewritten as according to the definition ( Polar cone definition ).

Proposition

(Local minimum of a sum). Let be a convex function, be a smooth function, be a subset of , be a local minimum of and let be convex. Then

Proof

The proof is a repetition of the proofs for the propositions ( Minimum of a smooth function ) and ( Minimum of a convex function ).

Optimality for smooth function figure 1

The figure ( Optimality for smooth function figure 1 ) illustrates the condition . The painted triangle is the constraint set . The ellipses are the level curves of a function with the internal ellipse is the level curve with the smallest value. The slightly transparent triangle is the set . The arrow is the vector . The is orthogonal to the level curve that passes through and points to the direction of increase of . The points in direction of decrease. Because the lies within the the point minimizes over . The alternative situation is presented on the picture ( Optimality for smooth function figure 2 ). Here, lies outside of the . In addition the must be orthogonal to the level curve. Therefore, the level curve must cross into thus preventing from being the minimum.

Optimality for smooth function figure 2

 Notation. Index. Contents.