Duality theorems

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Duality theorems Finding the dual optimal
solution from the primal optimal tableau
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Dual problem in Matrix form
In this lecture we shall present the primal and
dual problems in matrix form and prove certain
results on the feasible and optimal solutions of
the primal and dual problems. The material of
this lecture is treated in Chapter 7.5 of your
Textbook.
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Dual problem in Matrix form
Suppose the primal in (Matrix and ) standard form
is
Maximize
subject to
4
where
,
,
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Letting
the dual LPP in matrix form becomes
Minimize
subject to
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If the primal LPP is a minimization problem
Minimize
subject to
the dual LPP in matrix form becomes
Maximize
subject to
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Weak Duality Theorem For any pair of feasible
primal and dual solutions (X, Y), the value of
the objective function in the minimization
problem is an upper bound for the value of
objective function in the maximization problem.
For the optimal pair (X, Y), the values of the
objective functions of the two problems are equal.
Proof We first look at the case where the primal
is a maximization problem (so that the dual is a
minimization problem).
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Since X is a feasible solution of the primal,
(1)
and
(2)
Since Y is a feasible solution of the dual,
(3)
and
(4)
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Premultiplying the equation (2) by Y we get
Postmultiplying the inequality (4) by the non-
negative vector X we get
Thus we get
The proof when the primal is a minimization
problem is similar.
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Corollary(1) If X is a feasible solution of the
primal and Y is a feasible solution of the dual
such that
then X is an optimal solution of the primal and
Y is an optimal solution of the dual.
Remark The second part of the theorem says the
converse to the corollary is also true.
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Corollary(2) If one of the problems has an
unbounded solution then the other has an
infeasible solution.
For if it is not, then both problems have
feasible solutions and the relationship,
must hold, impossible as by assumption either
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The converse of this corollary is NOT true.
We can only say the following If one of the
problems has an infeasible solution, then the
other problem either has an unbounded solution or
has an infeasible solution.
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Example (1) Problem 2 Problem Set 7.5B
Page 325
Consider the LPP
Maximize
subject to
Show that the given primal problem is infeasible
and its dual is unbounded.
The primal is infeasible as 2nd constraint
violates
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The dual is the LPP
Minimize
subject to
is a feasible solution of the dual. Hence the
dual is unbounded.
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Example 2. Consider the Primal LPP
Maximize
subject to
Its dual is the LPP
Minimize
Both the problems have infeasible solutions as we
can easily show.
subject to
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x2
Primal
x1
y2
y1
Dual
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The following theorem tells us how we can find
the optimal solution of the dual from the optimal
tableau of the primal. Theorem Given the optimal
primal basis B and its associated objective
coefficient vector cB, the optimal solution of
the dual problem is
is often called the Simplex Multiplier.
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Proof Again we assume the primal is a
maximization problem. Since the table is optimal,
That is
or
So Y is feasible.
The optimal solution of the primal is
And hence by the corollary, Y is the optimal
solution of the dual.
Q.E.D.
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In words,
Row Vector of Optimal dual solution
( Row Vector of Original Objective coefficients
of optimal primal basic variables)
? ( optimal primal inverse )
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We can also say
Primal objective Row coefficient of variable xj
in the primal optimal tableau (LHS of
corresponding dual constraint) - (RHS of
corresponding dual constraint)
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Problem 4 Problem Set 7.5B Page 326
Consider the following LPP
Maximize
subject to

• Write the dual.
• Verify that B(A2,A3) is optimal.
• Find the associated optimal dual solution.

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Solution (b)
For non-basic variables x1, x4
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Also the primal solution is
Thus we get the optimal solution as
And Max z 16
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(a) The dual problem is
Minimize
subject to
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(c)
Thus the dual optimal solution is
y1 4, y2 0
We also note that Optimum w w
16 z