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.

## Minimax theory. et be a function where the and are subsets of and respectively. We always have Therefore, In this section we investigate the conditions for (Minimax equality)
and attainment of the sup and inf.

Definition

The pair is called a saddle point of iff for .

Proposition

(Saddle point's defining property). The pair is a saddle point iff the relationship ( Minimax equality ) holds and We introduce the function given by (definition of minmax p)

Proposition

(Minimax lemma 1).Assume that is convex for each . Then the function is convex.

Proof

The statement is a consequence of the propositions ( Preservation of convexity ) and ( Convexity of partial minimum ).

We will be using results of the section ( Min common and max crossing point section ). Following that section we define Proposition

(Minimax lemma 2). Let and is closed and convex for every . Then

1. 2. iff the relationship ( Minimax equality ) holds.

Proof

By definitions we have (q mu 1)
Since we do not increase the last quantity by choosing among the values. We obtain Next, we prove that when .

Take any small and fix . Since the function is convex then there is a separating hyperplane between the point and . Hence, the point lies below : and the lies above : We combine both inequalities into the statement where we claim existence of such for any . We transform the inequality as follows We intend to combine this result with the expression ( q_mu_1 ) above. Hence, we set and perform the operation . We obtain Hence, Next, we prove that when . Indeed, if then for any and any the point lies away from the epigraph of the convex function of . Hence, there is always a nonvertical hyperplane that separates any from and the lies in the upper half plane. Hence, where we claim existence of such and the statement holds for fixed and and any . Again, we apply the operation . Then the RHS becomes and the LHS may be let to . We conclude that With the representation proven we remark that and Therefore the statement (2) of the proposition follows.

Proposition

(Minimax theorem). Let and be nonempty convex subsets of and respectively and let be a function such that

1. For every the function is convex and closed,

2. For every the function is convex,

3. Then

1.The minimax equality holds iff the function given by the formula ( definition of minmax p ) is lower semicontinuous at 0.

2. If then the minimax equality holds and the supremum over Z in is finite and is attained. Furthermore, Proof

The statement follows from propositions ( Minimax lemma 1 )-( Minimax lemma 2 ) and ( Crossing theorem 1 )-( Crossing theorem 2 ) applied to the epigraph of p.

 Notation. Index. Contents.