Unformatted text preview: obtain
term outside the minimization, we obtain
{
If , then } { } { } which implies that
If , then }. Pulling the constant which implies that 203 { } In summary, we know that if
If
, then
, which is something that the outer optimization
problem would like to avoid (since it wants to maximize
). Therefore, we can express the Lagrangian dual as
D: maximize
subject to This can easily be rearranged into the form of the LP dual.
LPD: maximize
subject to Conclusion: In the case of LPs, the Lagrangian dual is equivalent to the standard LP dual. Structure of
We now return to the general problem P and its Lagrangian dual D.
P: minimize
subject to
D: maximize
subject to
where for ,
{ ∑ } { } Notice that for a given point
, the Lagrangian function is an affine function of u. For a
fixed
, we define this as follows. Then the function is looking for the smallest 204 over all We would construct one such line for each
. For a problem with linear constraints, if
we know that the optimal solution will occur at an extreme point, then we typically
construct one such line for each extreme point,
. Then the function
is
taken as the minimum over all
. (u ) hx3 (u ) hx 4 (u ) hx1 (u )
hx 2 (u ) u Theorem: The function is concave. Proof:
Select two distinct points ̂ ̅
we need to show that
̅
Applying the definition of
̅
Since , and let
̂ ̅ To show that
̂. , we have
{ ̂ ̂} ̅ is linear, it is both convex and concave. Therefore, we have
̅
and ̂ ̅ ̂ ̅ ̂ ̅ ̂, ̅ ̂ ̅ ̂ which implies that 205 is concave, Therefore, we have
{ ̂ ̅ { ̅ ̅ Aside: Note that for any two functions
{ ̂}
̂}
and } { , we have } { } Example: ̅ { ̂ { ̅ { Therefore, we have ̅} ̅ ̂ { ̅} ̅
Thus, we have shown that ̂}
̂}
{ ̂} ̂
̅ ̂ , so is concave. Example 2: Considering the following problem, derive its Lagrangian function
minimize
subject to This is such a simple problem that we can solve it intuitively.
= = In order to formulate the Lagrangian dual problem, we first put P into the given form.
206 P: minimize = subject to Therefore, its Lagrangian dual problem is given by
D: maximize
subject to ,
{ where ∑ } Note: For ease of notation, we will call the inner objective function
For a fixed value of ,
is a univariate function, so we can find its min explicitly and
therefore derive a closed form expression for
. To find the min, we set the first
derivative equal to 0 and verify that the second derivative is positive. Therefore, { } The Lagrangian dual problem is then given as
D: maximize
subject to
Again, due to the simplicity of this function, we can solve it easily. 207 This is verified by the following graph of . Optimal objective value for the Lagrangian dual problem (max value of
How does this compare to the optimal objective value of problem P? This simple problem illustrates the basic concepts of L...
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This document was uploaded on 11/28/2013.
 Fall '13

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