roblem
(Dual problem). Find
where
The dual problem delivers the highest crossing point for the set
Note that
is an
of a collection of affine functions. Hence, it is concave, upper
semi-continuous and may be studied with the means of the propositions
(
Crossing theorem
1
),(
Crossing theorem 2
). In particular,
the following statement directly follows from the proposition
(
Crossing theorem 1
), the geometrical
interpretation of the (
Visualization
lemma
) and the definition (
Geometric
multiplier
).
Proposition
(Duality gap and geometric
multipliers). The following alternative takes place.
1. If
(="there is no duality gap") then the set of geometric multipliers is equal to
the set of solutions of the problem (
Dual
problem
).
2. If
(="there is a duality gap") then the set of geometric multipliers is empty.
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