In which we introduce the theory of duality in linear programming.
1. The Dual of Linear Program
Suppose that we have the following linear program in maximization standard form:
and that an LP-solver has found for us the solution , , , of cost . How can we convince ourselves, or another user, that the solution is indeed optimal, without having to trace the steps of the computation of the algorithm?
Observe that if we have two valid inequalities
then we can deduce that the inequality
(derived by “summing the left hand sides and the right hand sides” of our original inequalities) is also true. In fact, we can also scale the inequalities by a positive multiplicative factor before adding them up, so for every non-negative values we also have
Going back to our linear program (1), we see that if we scale the first inequality by , add the second inequality, and then add the third inequality scaled by , we get that, for every that is feasible for (1),
And so, for every feasible , its cost is
meaning that a solution of cost is indeed optimal.
In general, how do we find a good choice of scaling factors for the inequalities, and what kind of upper bounds can we prove to the optimum?
Suppose that we have a maximization linear program in standard form.
For every choice of non-negative scaling factors , we can derive the inequality
which is true for every feasible solution to the linear program (2). We can rewrite the inequality as
So we get that a certain linear function of the is always at most a certain value, for every feasible . The trick is now to choose the so that the linear function of the for which we get an upper bound is, in turn, an upper bound to the cost function of . We can achieve this if we choose the such that
Clearly, we want to find the non-negative values such that the above upper bound is as strong as possible, that is we want to
So we find out that if we want to find the scaling factors that give us the best possible upper bound to the optimum of a linear program in standard maximization form, we end up with a new linear program, in standard minimization form.
Definition 1 If
is a linear program in maximization standard form, then its dual is the minimization linear program
So if we have a linear program in maximization linear form, which we are going to call the primal linear program, its dual is formed by having one variable for each constraint of the primal (not counting the non-negativity constraints of the primal variables), and having one constraint for each variable of the primal (plus the non-negative constraints of the dual variables); we change maximization to minimization, we switch the roles of the coefficients of the objective function and of the right-hand sides of the inequalities, and we take the transpose of the matrix of coefficients of the left-hand side of the inequalities.
The optimum of the dual is now an upper bound to the optimum of the primal.
How do we do the same thing but starting from a minimization linear program?
We can rewrite
in an equivalent way as
If we compute the dual of the above program we get
So we can form the dual of a linear program in minimization normal form in the same way in which we formed the dual in the maximization case:
- switch the type of optimization,
- introduce as many dual variables as the number of primal constraints (not counting the non-negativity constraints),
- define as many dual constraints (not counting the non-negativity constraints) as the number of primal variables.
- take the transpose of the matrix of coefficients of the left-hand side of the inequality,
- switch the roles of the vector of coefficients in the objective function and the vector of right-hand sides in the inequalities.
Fact 2 The dual of the dual of a linear program is the linear program itself.
We have already proved the following:
Fact 3 If the primal (in maximization standard form) and the dual (in minimization standard form) are both feasible, then
Which we can generalize a little
Theorem 4 (Weak Duality Theorem) If is a linear program in maximization standard form, is a linear program in minimization standard form, and and are duals of each other then:
- If is unbounded, then is infeasible;
- If is unbounded, then is infeasible;
- If and are both feasible and bounded, then
Proof: We have proved the third statement already. Now observe that the third statement is also saying that if and are both feasible, then they have to both be bounded, because every feasible solution to gives a finite upper bound to the optimum of (which then cannot be ) and every feasible solution to gives a finite lower bound to the optimum of (which then cannot be ).
What is surprising is that, for bounded and feasible linear programs, there is always a dual solution that certifies the exact value of the optimum.
Theorem 5 (Strong Duality) If either or is feasible and bounded, then so is the other, and
To summarize, the following cases can arise:
- If one of or is feasible and bounded, then so is the other;
- If one of or is unbounded, then the other is infeasible;
- If one of or is infeasible, then the other cannot be feasible and bounded, that is, the other is going to be either infeasible or unbounded. Either case can happen.
We will return to the Strong Duality Theorem, and discuss its proof, later in the course.