Lecture Notes Discrete Optimization - Applied Mathematics

largest lower bound onOPT LP ) corresponds to the so-called dual program of (2).
maximize
subject to
m
∑
i=1
m
∑
i=1
b i y i
a i j y i ≤ c j ∀ j ∈{1,...,n}
y i ≥ 0 ∀i∈{1,...,m}
(4)
We use OPT DP to refer to the objective function value of an optimal solution to the dual
linear program (4).
There is a strong relation between the primal LP (2) and its corresponding dual LP (4).
Note that (3) shows that the objective function value of an arbitrary feasible dual solution
(y i ) is less than or equal to the objective function value of an arbitrary feasible primal
solution (x j ). In particular, this relation also holds for the optimal solutions and thus
OPT DP ≤ OPT LP . This is sometimes called weak duality. From linear programming
theory, we know that even a stronger relation holds:
Theorem 1.1 (strong duality). Let x=(x j ) and y=(y i ) be feasible solutions to the LPs
(2) and (4), respectively. Then x and y are optimal solutions if and only if
n
∑
j=1
c j x j =
m
∑
i=1
b i y i .
An alternative characterization is given by the complementary slackness conditions:
Theorem 1.2. Let x = (x j ) and y = (y i ) be feasible solutions to the LPs (2) and (4),
respectively. Then x and y are optimal solutions if and only if the following conditions
hold:
1. Primal complementary slackness conditions: for every j∈{1,...,n}, either x j = 0
or the corresponding dual constraint is tight, i.e.,
∀ j ∈{1,...,n} : x j > 0 ⇒
m
∑
i=1
a i j y i = c j .
2. Dual complementary slackness conditions: for every i∈{1,...,m}, either y i = 0
or the corresponding primal constraint is tight, i.e.,
∀i∈{1,...,m} : y i > 0 ⇒
n
∑
j=1
a i j x j = b i .
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