Rockafellar, Part VII: Saddle-Functions and Minimax Theory

Sections 33–37: concave-convex functions and their two partial closures, the equivalence classes of closed saddle-functions, continuity and differentiability, minimax problems, and the conjugacy correspondence that carries the existence theory of saddle-values. This module imports the five section modules and adds nothing of its own. All 58 numbered results of Part VII are formalized.

§	module	subject
33	Part7.Section33	Saddle-Functions
34	Part7.Section34	Closures and Equivalence Classes
35	Part7.Section35	Continuity and Differentiability of Saddle-Functions
36	Part7.Section36	Minimax Problems
37	Part7.Section37	Conjugate Saddle-Functions and Minimax Theorems

Three orientation conventions run through the Part and will invert statements if misread: cl₁ closes the concave — first — argument and cl₂ the convex — second (§33); from §36 onward minimisation takes place in the convex argument and maximisation in the concave; and the lower conjugate is sup_v inf_u while the upper is inf_u sup_v (§37). Each is stated where it is introduced.

References

• R. T. Rockafellar, Convex Analysis, Princeton University Press, 1970, §§33–37.