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DUALITY. IENG511 Optimization Theory Assis.Pro.Dr.Sahand Daneshvar Ramin Soufi Spring 2013-14. DUALITY. Associated with each linear programming problem there is another linear programming problem called the dual.
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DUALITY IENG511 Optimization Theory Assis.Pro.Dr.SahandDaneshvar RaminSoufi Spring 2013-14
DUALITY Associated with each linear programming problem there is another linear programming problem called the dual. The dual linear program possesses many important properties relative to the original primal linear program. There are two important forms (definitions) of duality: the canonical form of duality and the standard form of duality. These two forms are completely equivalent. They arise respectively from the canonical and the standard representation of linear programming problems.
Canonical Form of Duality Suppose that the primal linear program is given in the (canonical) form: P : Minimize cx subject to Ax > b x > 0. Then the dual linear program is defined by: D : Maximize wb subject to wA < c w > 0. Note that there is exactly one dual variable for each primal constraint and exactly one dual constraint for each primal variable.
Example: Consider the following linear program and its dual:
Standard Form of Duality Another equivalent definition of duality may be given with the primal linear program stated in the following standard form: P : Minimize cx subject to Ax = b x > 0. Then the dual linear program is defined by: D : Maximize wb subject to wA < c w unrestricted.
But since -wl < 0 is the same as w > 0, we obtain the canonical form of the dual problem. Given one of the definitions, canonical or standard, it is easy to demonstrate that the other definition is valid. For example, suppose that we accept the standard form as a definition and wish to demonstrate that the canonical form is correct. By adding slack variables to the canonical form of a linear program, we may apply the standard form of duality to obtain the dual problem.
Dual of the Dual Since the dual linear program is itself a linear program, we may wonder what its dual might be. Consider the dual in canonical form: Maximize wb subject to wA < c w > 0. Applying the transformation techniques of Chapter 1, we may rewrite this problem in the form: Minimize (-b' )w' subject to (-A')w' > (-c') w' > 0.
The dual linear program for this linear program is given by (letting x' play the role of the row vector of dual variables): Maximize x‘ (-c' ) subject to x‘ (-A') < (-b') x' > 0. But this is the same as: P : Minimize cx subject to Ax > b x > 0, which is precisely the original primal problem. Thus, we have the following lemma, which is known as the involutory property of duality.
