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Unformatted text preview: he dual of a problem P, then the dual of D is P. Informally, the dual of the
dual is the primal. 9.3.3 Duality Theorem An important aspect of duality is that feasible solutions of the primal and the dual are related.
Lemma 9.8. Any feasible solution of Problem 9.6 yields an upper bound for Problem 9.5. In
other words, the value given by any feasible solution of the dual of a problem is an upper bound
for the primal problem.
Proof. Let (y1 , . . . , ym ) be a feasible solution of the dual and (x1 , . . . , xn ) be a feasible solution
of the primal. Then,
n n j=1 j= ∑ c jx j ≤ ∑ m m i= i=1 ∑ ai j yi x j ≤ ∑ n m j=1 i=1 ∑ ai j x j yi ≤ ∑ biyi. Corollary 9.9. If (y1 , . . . , ym ) is a feasible solution of the dual of a problem (Problem 9.6)
and (x1 , . . . , xn ) is a feasible solution of the corresponding primal (Problem 9.5) such that
∑n=1 c j x j = ∑m 1 bi yi , then both solutions are optimal.
j
i=
Corollary 9.9 states that if we ﬁnd two solutions for the dual and the primal achieving the
same value, then this is a certiﬁcate of the optimality of these solutions. In particular, in that
case (if they are feasible), both the primal and the dual problems have same optimal value.
For instance, we can easily verify that (0, 14, 0, 5) is a feasible solution for Problem 9.4 with
value 29. On the other hand, (11, 0, 6) is a feasible solution for the dual with same value. Hence,
the optimal solutions for the primal and for the dual coincide and are equal to 29.
In general, it is not immediate that any linear programme may have such certiﬁcate of optimality. In other words, for any feasible linear programme, can we ﬁnd a solution of the primal
problem and a solution of the dual problem that achieve the same value (thus, this value would
be optimal)? One of the most important result of the linear programming is the duality theorem
that states that it is actually always the case: for any feasible linear programme, the primal and
the dual problems have the same optimal solution. This theorem has been proved by D. Gale,
H.W. Kuhn and A. W. Tucker [5] and comes from discussions between G.B. Dantzig and J. von
Neumann during Fall 1947.
Theorem 9.10 (D UALITY T HEOREM ). If the primal problem deﬁned by Problem 9.5 admits an
∗
∗
optimal solution (x1 , . . . , xn ), then the dual problem (Problem 9.6) admits an optimal solution
(y∗ , . . . , y∗ ), and
m
1
n m j=1 i=1 ∑ c j x∗j = ∑ biy∗.
i 142 CHAPTER 9. LINEAR PROGRAMMING Proof. The proof consists in showing how a feasible solution (y∗ , . . . , y∗ ) of the dual can be
m
1
obtained thanks to the Simplex Method, so that z∗ = ∑m 1 bi y∗ is the optimal value of the primal.
i=
i
The result then follows from Lemma 9.8.
Let us assume that the primal problem has been solved by the Simplex Method. For this
purpose, the slack variables have been deﬁned by
n xn+i = bi − ∑ ai j x j for 1 ≤ i ≤ m.
j=1 Moreover, the last line of the last dictionary computed during...
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This note was uploaded on 11/20/2013 for the course CS 101 taught by Professor Smith during the Fall '13 term at Mitchell Technical Institute.
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