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.

## Existence of solution of convex optimization problem.

roposition

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

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

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

Proof

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

Proposition

(Basic existence result). Let be a closed convex subset of and let be a closed proper convex function such that . The set is nonempty and compact if and only if the and have no common directions of recession.

If and 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 with the sequence converging to the . 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 be a closed convex subset of and let be a closed proper convex function such that . The set is nonempty if any of the following conditions hold.

1. .

2. and is given by the linear constraints for some .

3. and are of the form where the are positive semidefinite matrixes.

Remark

The convex function is constant on the subspace .

 Notation. Index. Contents.