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4.1. LINEAR OPTIMIZATION 21
be a primal-dual feasible solution, and let
(x c ) ∗ := Ax ∗ .
For a primal-dual feasible solution we define the duality gap as the difference between the primal and
the dual objective value,
=
m−1
∑
i=0
c T x ∗ + c f − ( (l c ) T (s c l ) ∗ − (u c ) T (s c u) ∗ + (l x ) T (s x l ) ∗ − (u x ) T (s x u) ∗ + c f )
n−1
∑
[(s c l ) ∗ i ((x c i) ∗ − li c ) + (s c u) ∗ i (u c i − (x c i) ∗ [
)] + (s
x
l ) ∗ j (x j − lj x ) + (s x u) ∗ j (u x j − x ∗ j ) ]
≥ 0
j=0
(4.3)
where the first relation can be obtained by transposing and multiplying the dual constraints (4.2) by
x ∗ and (x c ) ∗ respectively, and the second relation comes from the fact that each term in each sum
is nonnegative. It follows that the primal objective will always be greater than or equal to the dual
objective.
4.1.1.3 When the objective is to be maximized
When the objective sense of problem (4.1) is maximization, i.e.
maximize
c T x + c f
subject to l c ≤ Ax ≤ u c ,
l x ≤ x ≤ u x ,
the objective sense of the dual problem changes to minimization, and the domain of all dual variables
changes sign in comparison to (4.2). The dual problem thus takes the form
minimize (l c ) T s c l − (u c ) T s c u + (l x ) T s x l − (u x ) T s x u + c f
subject to A T y + s x l − s x u = c,
− y + s c l − s c u = 0,
s c l , s c u, s x l , s x u ≤ 0.
This means that the duality gap, defined in (4.3) as the primal minus the dual objective value, becomes
nonpositive. It follows that the dual objective will always be greater than or equal to the primal
objective.
4.1.1.4 An optimal solution
It is well-known that a linear optimization problem has an optimal solution if and only if there exist
feasible primal and dual solutions so that the duality gap is zero, or, equivalently, that the complementarity
conditions