Topic 19 - Intro to Lagrangian Duality

re arranging terms we obtain term outside the

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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 re-arranged 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.

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