- 1
minimize
subject to ,
where is a nonempty set,
and
and
, are given functions.
: optimal value.
feasible (vecter) : satisfying the CP
- 2
maximize
subject to
where the dual function is:
and is the Lagrangian function defined by
,
;
the effective constraint set of the dual problem is :
: optimal value of the DP.
weak duality relation :
as well as:
for all , and
with
,
so that:
we state this formally as follows.
- Proposition
1.Deak Duality Theorem:
For any feasible solution and any
, we have
. Moreover,
.
2.Optimality Conditions:
There holds , and
are a primal and dual optimal solution pair if and only if
is feasible,
, and
.
3.Strong Duality -Existence of Dual optimal solutions:
Assume the set is convex, and the functions
, and
, ...,
are convex. Assume further that
is finite, and that one of the following twoconditions holds:
(1). There exists such that
for all
.
(2). The functions , are affine, and there exists
such that
.
Then are there exists at least one dual optimal solution. Under condition (1) the set of dual optimal solutions is also compact.