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Viterbi School of Engineering
Daniel J. Epstein Department of Industrial and Systems Engineering
ISE 330: Introduction to Operations Research
Fall 2006 (Oct 2): Duality Theory
http://www-scf.usc.edu/~ise330
Read: H&L chapter 6.3-6.4
Review: Duality Theory
Weak duality property: If x is a feasible solution for the primal problem and y is a feasible
solution for the dual problem, then cx ≤ yb.
Strong duality property: If x* is an optimal solution for the primal problem and y* is an
optimal solution for the dual problem, then cx* = y*b.
Complementary solutions property: At each iteration, the simplex method simultaneously
identifies a CPF solution x for the primal "roblem and a complementary
solution y for the dual problem (found in row 0, the coeff of the slack
variables), where cx = yb. If x is not optimal for the primal, y is infeasible.
Complementary optimal solutions property: At the 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 (found in row 0,
the coeff of the slack variables), where cx* = y*b. In this solution, y* gives
the shadow prices for the primal problem.
Symmetry property: The dual of the dual is the primal.
Find the corresponding complementary basic solutions based on the simplex tableaux:
Final
Basic
Coeff
Tableaux
Var
Z
original
slack
RHS
Z
1
cBB-1A – c
cBB-1
cBB-1b
xB
0
B-1A
B-1
B-1b
