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Topic 19 - Intro to Lagrangian Duality

# Topic 19 - Intro to Lagrangian Duality - ISE 5406 Topic#19...

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201 ISE 5406 Topic #19: Intro to Lagrangian Duality BSS Textbook Reading: Sections 6.1 6.3 We now give a brief introduction to the topic of Lagrangian Duality (presented in detail in Chapter 6 of the BSS textbook). Lagrangian duality is a powerful optimization technique that is useful for nonlinear programs, integer programs, and large-scale linear programs. The time-frame of our course does not allow us to cover this material in depth, but the hope is that you will have enough understanding of Lagrangian duality to be able to seek out further resources if you need to implement it in the future. Problem Background Suppose that we are addressing the following problem, where . P: minimize subject to for Note: Chapter 6 of the BSS textbook derives the Lagrangian dual for a problem that also contains equality constraints. For simplicity, we concentrate on a problem with only inequalities, but the extensions are natural and clearly presented in the text. Then the Lagrangian dual is given by D: maximize subject to , where { ∑ } { } Notes: is a vector function. That is, with ( ) . Technically, the outer “maximization” problem should seek the supremum, since a max may not exist. Similarly, the inner optimization only seeks an inf, but frequently I am sloppy with notation and use “min” instead. It is understood that an actual min may not exist, and an inf is all that is required. For a given problem P, there can be more than one Lagrangian dual formulation, depending upon what constraints are included in X .

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202 Example 1: Provide two different Lagrangian dual formulations for the following problem by using the definitions of X provided. Minimize subject to Formulation #1: Define { } D1: maximize subject to , where Formulation #2: Define { } D2: maximize subject to , where In many cases, the decision about which constraints to include in the set has a significant impact on the solvability of the Lagrangian dual. There is often a trade-off between the ease of solving the dual and the value of the bound it provides for the primal. Connection to LP Duality How does this compare to LP duality? Consider an LP primal-dual pair in canonical form. LP: minimize LPD: maximize subject to subject to
203 Recall that in order to formulate the Lagrangian dual, we need to have our problem in the following form. P: minimize subject to We can put LP into this form as follows. Therefore, the Lagrangian dual would be given as D: maximize subject to , where { } Conceptually: Outer optimization problem: Selects values for u . Inner optimization problem: With the u values fixed, selects the values of x that minimize the inner objective function. Then the outer problem tries to find a better value of u to maximize , and the process is repeated. Re-arranging terms, we obtain { } . Pulling the constant term outside the minimization, we obtain { } { } If , then { } which implies that If , then { } which implies that

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204 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 ).
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Topic 19 - Intro to Lagrangian Duality - ISE 5406 Topic#19...

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