computational methods for management and economics carla gomes module 7a duality
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Computational Methods forManagement and Economics
Carla Gomes
Module 7aDuality
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Duality
• Every maximization LP problem in the standard form gives rise to a minimization LP problem called the dual problem
• Every feasible solution in one yields a bound on the optimal value of the other
• If one of the problems has an optimal solution, so does the other and the two optimal values coincide
• These results have very interesting economic interpretations
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Bounds on the optimal value
Let x1 = the number of doors to producex2= the number of windows to produce
Maximize Z= 3 x1 + 5 x2subject to
x1≤ 42x2≤ 123x1 + 2x2≤ 18
andx1 ≥ 0, x2 ≥ 0.
Lower bound (LB) on Z* Z* ≥ LB any feasible solutionUpper bound (UB) on Z* Z* ≤ UB – how do we come up with
upper bounds for Z* for a maximization LP problem?
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“Guessing” upper bounds
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A principled way of finding upper-bounds on Z*
Dual Problem
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Primal vs. Dual Problem
Maximize Z= 3 x1 + 5 x2
subject to
x1 ≤ 4 2x2≤ 123x1 + 2x2≤ 18
and
x1 ≥ 0, x2 ≥ 0.
Minimize W= 4 y1 + 12 y2 + 12 y3
subject to
y1 + 3 y3 ≥ 3 2 y2 + 2 y3 ≥ 5
and
y1 ≥ 0, y2 ≥ 0, y3 ≥ 0
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Primal vs. dual
• The dual of a maximization problem is a minimization problem; the dual of a minimization problem is a maximization problem
• The m primal constraints are in a one-to-one correspondence with the m dual variables yi, and conversely, the n dual constraints are in a one-to-one correspondence with the n primal variables, xj.
• The coefficient at each variable in the objective function, primal or dual, appears in the other problem the RHS of the corresponding constraint.
ibjxija
jciyija
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Primal vs. Dual
Max Z = c1 x1 + c2x2 + …. + cnxn
Subject to:
a11 x1 + a12 x2 + … + a1n xn ≤ b1
a21 x1 + a22 x2 + … + a2n xn ≤ b2
…
am1 x1 + am2 x2 + … + amn xn ≤ bmx1, x2, …, xn ≥ 0
Min W = b1 y1 + b2y2 + …. + bmym
Subject to:
a11 y1 + a21 y2 + … + am1 ym ≥ c1
a12 y1 + a22 y2 + … + am2 ym ≥ c2
…
a1n y1 + a2n y2 + … + amn yn ≥ cny1, y2, …, ym ≥ 0
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Primal vs. Dual
Max Z = c x
Ax ≤ bx ≥ 0
Max W = bT y
ATy ≤ cT
y ≥ 0
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Primal vs. Dual Problem
Maximize Z= 3 x1 + 5 x2
subject to
1 x1 ≤ 4 2 x2≤ 123 x1 + 2 x2≤ 18
and
x1 ≥ 0, x2 ≥ 0.
Minimize W= 4 y1 + 12 y2 + 18 y3
subject to
1 y1 + 3 y3 ≥ 3 2 y2 + 2 y3 ≥ 5
and
y1 ≥ 0, y2 ≥ 0, y3 ≥ 0
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Wyndor Glass: Primal vs. DualMatrix Notation
Max Z = 3 5 x1
x2
Subject to:
1 0
0 1
3 2
4
12
18
x1
x2
≤
x ≥ 0
Min W= 4 12 18 y1
y2
y3
1 0 3 0 1 2
y1y2y3
≥35
y ≥ 0
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Primal-dual table for LP
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Max
n
jj
xj
c1
s.t
),,2,1(1
miibjxn
j ija
),,2,1(0 njjx
m
ii
yi
b1
Min
),,2,1(1
njjciym
i ija
),,2,1(0 miiy
Primal Dual
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Every feasible dual solution yields an upper-bound on Z* of the primal and every feasible primal solution yields a lower-
bound on Z* of the dual
Z*
Primal feasible solutions(Maximization)
Lower Bounds on Dual Optimum
Dual feasible solutions(Minimization)
Upper Bounds on Primal Optimum
Weak duality property
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m
ii
yi
bn
jj
xj
c11
Weak duality property
Weak duality property
Every feasible dual solution yields an upper-bound on Z* of the primal and every feasible primal solution yields a lower-bound on Z* of the dual
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Proof: Weak duality property
m
ii
yi
bn
jj
xj
c11
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Primal vs. dual Simplex
My-wyndor.txtMy-wyndor-dual.txt
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The magic continues …Final tableau - primal
Final tableau - dual
Final row (0) of the
simplex automatically
provides the optimal
values of the
dual variables, y*i ,
they correspond to
the slack variables in
the final row(0) of the
simplex method.
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Strong Duality Property
m
ii
yi
bn
jj
xj
c1
*
1
*
For the proof we’ll use as starting point the final row (0)as generated by the algebraic simplex procedure. We’llalso show how the final row (0) of the simplex algebraicprocedure automatically provides the optimal values of thedual variables. In fact we denote by y*
i the coefficients that correspond to the slack variables in the final row(0) of the simplex method.
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Every feasible dual solution yields an upper-bound on Z* of the primal and every feasible primal solution yields a lower-
bound on Z* of the dual
Z*
Primal feasible solutions(Maximization)
Lower Bounds on Dual Optimum
Dual feasible solutions(Minimization)
Upper Bounds on Primal Optimum
Weak duality property
Strong duality property
Primal Opt. = Dual Opt.
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Strong Duality
Property Proof
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Notation for entries in row 0 of a simplex tableau (textbook)
Dual surplus variables Dual decision variables
Note: Complete basic solutions for the primal and dual problem can be read directly from row (0)
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Minimize W= 4 y1 + 12 y2 + 12 y3
subject to
y1 + 3 y3 ≥ 3 12 y2 + 12 y3 ≥ 5
and
y1 ≥ 0, y2 ≥ 0, y3 ≥ 0
Minimize W= 4 y1 + 12 y2 + 12 y3
subject to
y1 + 3 y3 – (z1 – c1) = 3
12 y2 + 12 y3 - (z2 – c2) = 5
and
y1, y2 , y3 , (z1 – c1), (z2 – c2) ≥ 0
Introducingsurplus vars
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Primal Simplex: Weak Duality and strong dualityThe different (non-optimal) basic feasible solutions produced by the
simplex algorithm in each iteration are feasible from the point of view
of the primal but not from the point of view of the dual.
Iteration 0: basic dual solution is infeasible since(z1 – c1) and (z2 – c2) are negative (weak duality)
Iteration 1: basic dual solution is infeasible because (z1 – c1) is negative (weak duality)Final iteration –basic dual solution is feasible and optimal (StrongDuality).
Note: Complete basic solutions for the primal and dual problem can be read directly from row (0)
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Complementary slackness relationship for complementary basic solutions
Primal Variable Associated Dual Variable
Basic
Non-Basic
Non-basic (m variables)
Basic (n variables)
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Simplex method:Complementary solutions property
• At each iteration the simplex method simultaneously identifies a CPF solution x for the primal problem and a complementary solution y for the dual problem, where:– cx = yb– If x is not optimal for the primal, than y is not
feasible for the dual.
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Simplex method:Complementary optimal solutions
property
• At final iteration the simplex method simultaneously identifies an optimal solution x* for the primal problem and a complementary optimal solution y* for the dual problem, where:– cx* = yb*
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Complementary Slackness Optimal solutions
• Let x1*, x2*, …xn* be a primal feasible solution and a y1*, y2*, …ym* dual feasible solution.
• Necessary and sufficient conditions for simultaneous optimality of x1*, x2*, …xn* and y1*, y2*, …ym* are:
),,2,1()(0*
1
* njbothorjxorjcm
ii
yi
b
),,2,1()(0*
1
* mibothoriyoribn
jj
xij
a
This result is very useful for checking the optimality of solutions
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Complementary Slackness Optimal solutions
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Relationships between the primal and dual problems
• If one problem has feasible solutions and a bounded objective function (and therefore an optimal solution), then so does the other problem (both weak and strong duality apply)
• If one problem has feasible solutions and an unbounded objective function (therefore no optimal solution) then the other problem has no feasible solutions
• If one problem has no feasible solutions than the other problem has no feasible solutions or an unbounded objective function.
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Primal-Dual Combinations
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Adapting to Other Forms
• What if our problem is not in standard form?
– We can always transform it to the standard form and then construct the dual;
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Dual of the dual
Note: it is not important which problem we call dual and whichproblem we call primal given the symmetry property of the primaldual relationships. In general we call primal the model formulatedto fit the actual problem.
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Shortcut for = constraints
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Shortcut for dual transformations: = constraints or unconstrained variables
We could still do the conversion into the standard format ,but a shortcut is available for these two formats.
•An equality constraint in the primal should be treated justlike a <= constraint in constructing the dual, except that the non-negativity constraint for the corresponding dual variableshould be deleted•By the symmetry property, a primal variable not subject tonon-negativity constraints affects the dual problem only by changing the corresponding inequality constraint to anequality constraint.
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Shortcut for dual transformations: >= constraints (Maximization)
Consider:
•Constructing the dual from this expression would lead to –aij ascoefficients of yi in the functional constraint j (of the form >=)and a coefficient –bi in the objective function (which is to beminimized); yi would be subject to a non-negativity constraint (yi ≥ 0);•Cleaner alternative for dual:
•Consider y’i = - yi •Express the dual in terms of y’i
(- the coefficients of the variable become aij for functional constraint j and bi for the objective function- the constraint in the variable becomes y’i ≤ 0;)
ibjxn
j ijaibjxn
j ija
11
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Sensible, Odd, Byzarre Variable constraints
• Maximization/Minimization problems
• “sensible” x >=0;• “odd” x unrestricted;• “byzarre” x <=0;
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Sensible, Odd, Byzarre Constraints
• Maximization problems• “sensible” <=
• “odd” =
• “byzarre” >=
• Minimization problems• “sensible” >=
• “odd” =
• “byzarre” <=
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Radiation Therapy Example
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Primal-dual correspondence
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Complexity of Simplex Method Primal vs. Dual
• How long does it take to solve an LP using the simplex method? – Several factors but the most important one seems to be the
number of functional constraints.• Computation tends to be proportional to the cube of the number of
functional constraints in an LP.
– The number of variables is a relatively minor factor (assuming revised simplex method)
– The density of the matrix of technological coefficients is also a factor – the sparser the matrix (i.e., the larger the number of zeroes) the faster the simplex method;
Real world problems tend to be sparse, i.e., “sparcity” of 5% or even 1%, which leads to fast runs.
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Complexity of Simplex Method Primal vs. Dual
• But the most important one seems to be the number of functional constraints.– Computation tends to be proportional to the
cube of the number of functional constraints in an LP.
Problem A takes 8 times
longer than problem B
Question: So if problem A has twice as many
constraints as problem B how much longer takes
to solve problem A in comparison to problem B?
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Primal vs. Dual?
So, the size of the problem, may determinewhether to use the simplex method on the
primal or dual problem.
If the primal has a large number of constraints and a small number of variables it is better to apply
the simplex method to the dual (since it will havea small number of constraints).
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Dual Simplex Method
• This method is based on the duality results.• It is a mirror image of the simplex method:
– the simplex method deals with primal feasible solutions (but not dual feasible), moving toward a solution that is dual feasible;
– the dual method deals with basic solutions in the primal problem that are dual feasible but not primal feasible. It moves toward an optimal solution by striving to achieve primal feasibility as well.