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.

## Partial minimization of convex functions.

roposition

(Convexity of partial minimum). Let be a convex function. Then the function given by is convex.

The proof of the above proposition is a direct verification based on definitions.

The study of closeness of the partial minimum is based on the following observation.

Suppose the level set is nonempty for some . Let be a sequence such that . Then The set is closed if is closed. The set is a projection of . Its closeness may be studied by means of the proposition ( Preservation of closeness result ). The intersection preserves the closeness. Hence, we arrive to the following proposition.

Proposition

(Partial minimization result). Let be a closed proper convex function. Then the function is closed, convex and proper if any of the following conditions hold.

0. There exist and such that is nonempty and compact.

1. There exist and such that is nonempty and .

2. ,for some , where the is given by the linearity constraints and there exists such that

3. , where the is given by the quadratic constraints where the are positive semidefinite and there exists such that

 Notation. Index. Contents.