Unformatted text preview: agrangian duality. is a concave function
is the minimum (actually the infimum) of a number of linear functions (one for
each
).
In this example, we saw that the primal and dual problems had equal objective
values, just like in LPs. (It remains to be seen whether or not that is always the case
for nonlinear problems.) To reinforce concepts, we provide another example of how the Lagrangian dual can be used
to provide an analytical solution. This time, we’ll address a problem with two variables. Example 3: Formulate and solve the Lagrangian dual for the following primal problem.
Minimize
subject to 208 Let us define
D: maximize
subject to { }. Then, the Lagrangian dual is given by , where As before, for a fixed value of , we can refer to the inner objective as . That is, We can search for the unconstrained min of this function by setting its gradient equal to
zero. Therefore, we find that a candidate solution for the minimum is given by We note that this solution is always feasible to the constraints
{
} for any
. We can verify that this solution is a minimum by checking that the
Hessian matrix is PSD (and hence p is convex) for all x. Now that we have closed form values for the optimal x, we can restate the Lagrangian dual
as follows.
D: maximize
subject to
Substituting our optimal values (
D: maximize () ), we obtain
( )( ) subject to
209 () ( () ( )) Therefore, we have the following graph of . We can find the unconstrained max of this function by setting its derivative equal to zero. So our candidate solution for the max is
To verify that this is a max, we check and see that Therefore, we have found a global max at This corresponds to a maximal value of How does this relate to the primal problem? Substituting for the optimal values of
, we obtain and the associated primal objective value is 210 and This is the second example we’ve seen of Lagrangian dual problems that could be solved
analytically, and in both cases, we saw that the optimal objective values for the primal and
Lagrangian dual problems were equal. We now discuss the objective values of the two
problems in more detail. Geometric Interpretation of the Primal and Lagrangian Dual Objectives
In order to see the relationship between the primal and Lagrangian dual objective values, it
is instructive to map the points in
based upon their values of
and
. In order
to keep things simpler, we will consider a problem with a single (inequality) constraint.
Therefore, the primal problem we will consider is
P: minimize
subject to We want to map each feasible point to a place in the
interested in the set G, where plane. Specifically, we are { } Example 4: Recall the problem of Example 2, reproduced below.
minimize
subject to In this case, we would take Rearranging terms, we would have We can now...
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This document was uploaded on 11/28/2013.
 Fall '13

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