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25 votes
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What's the difference between Lagrangian relaxation and Lagrangian decomposition?

They are not the same thing. Lagrangian decomposition is a special case of Lagrangian relaxation. (Note: I'm talking specifically about integer programming problems in this answer, though some of ...
LarrySnyder610's user avatar
9 votes
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"Partial" Lagrangian Dual in LP

Based on the mentioned references, suppose the primal problem is: \begin{align} \begin{array}{cl} \underset{}{\text{minimize}} & c x \\ \text{subject to} & Ax = a \\ & Dx \leq e \\ & x ...
A.Omidi's user avatar
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9 votes

"Partial" Lagrangian Dual in LP

This is called Lagrangian relaxation, no matter what subset of constraints you choose to dualize.
RobPratt's user avatar
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6 votes
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Difficulties with Finding a Proper Penalty Value for the Progressive Hedging Algorithm

This problem is addressed in some detail in Section 2.1 of the paper Progressive hedging innovations for a class of stochastic mixed-integer resource allocation problems by Watson and Woodruff (a non-...
David M.'s user avatar
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5 votes
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Augmented Lagrangian Function for Semidefinite Programming Problems

My way of reading it is $\langle X, \mathcal{A}^*(y)+S-C\rangle = \langle X, \mathcal{A}^*(y)-C\rangle + \langle X,S \rangle$. The first term is your standard inner product between dual variable and ...
Johan Löfberg's user avatar
5 votes
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Lagrangian Relaxation bound greater than optimal solution

After the discussion here and a suggestion on my post at gurobi's community I'll post an answer for the forum records. Concerning the presolve, I found out that in order to check if this is what is ...
seimetz's user avatar
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5 votes
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How to Speed Up the subgradient optimization procedure in a Lagrangian Relaxation Scheme

In my experience, this sort of thing just requires a lot of trial and error. An SO variant that works well for one problem may not work as well for another, so you just have to implement and test a ...
LarrySnyder610's user avatar
5 votes

Using LR-based method to solve mixed integer programming

As its name indicates, a Lagrangian relaxation is a relaxation and therefore only provides a dual bound. If you are interested in getting a primal solution, you have several ways to exploit a ...
fontanf's user avatar
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4 votes

Lagrangian Relaxation The Weak Lower Bound

A problem with subgradient optimization is the phenomenon of “zigzagging” in the vicinity of the optimal solution. That is, it is often seen that the direction of search in one step is almost opposite ...
Sune's user avatar
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4 votes

Lagrangian Relaxation for Two-Stage Stochastic Program

I'll comment on the Lagrangian relaxation question and leave the Benders question for someone else to comment on. (You might want to consider splitting your question into two, one for LR and one for ...
LarrySnyder610's user avatar
4 votes

The variable splitting scheme in the context of Lagrangian relaxation

I am familiar with variable splitting (also known as Lagrangian decomposition) being applied to facility location problems. We used it in our paper: Snyder and Daskin, Stochastic $p$-robust location ...
LarrySnyder610's user avatar
4 votes

The variable splitting scheme in the context of Lagrangian relaxation

The only occurrences of this decomposition that I am aware of are from: Jörnsten K, Näsberg M (1986) A new Lagrangian relaxation approach to the generalized assignment problem. European Journal of ...
fontanf's user avatar
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4 votes

Augmented Lagrangian Function for Semidefinite Programming Problems

There's a good discussion of this in Convex Optimization by Stephen Boyd and Lieven Vandenberghe. See section 5.9. With an ordinary scalar inequality constraint: $f_{i}(x) \leq 0$, you'll have a term ...
Brian Borchers's user avatar
3 votes

Lagrangian Relaxation Lower Bound exceeds the Upper bound and the Optimal solution

That sounds like a good choice for a relaxation, but you haven't explicitly shown how you compute the bounds. From your description so far, I suspect that you have two errors. First, the objective ...
RobPratt's user avatar
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3 votes
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Related to Lagrangian dual

Dualizing a constraint comes back to the first, the direction of the objective function, and the second, how the dualized constraint would be violated. In your case, the constraint is written as $LHS-...
A.Omidi's user avatar
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2 votes

Lagrangian Relaxation for Two-Stage Stochastic Program

Larry Snyder explained very well. Few items to check/add for the Lagrangean Relaxaiton part: Make sure your lowerbounds- i.e. the feasible solution (most probably depending on fixed first stage ...
Evren Guney's user avatar
2 votes

Partial Lagrangian in the Max-Flow problem

Unfortunately, there is NO guarantee to find the primal feasible solution in the context of Lagrangian relaxation, especially when it comes to solving integer programming. In order to get a feasible ...
A.Omidi's user avatar
  • 8,940
2 votes
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Partial Lagrangian in the Max-Flow problem

The error comes in assuming that, given a zero duality gap, the solution to the relaxed LP (which I agree becomes a shortest path problem) would be the optimal flow solution. Let $X$ the set of all ...
prubin's user avatar
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2 votes

Related to Lagrangian dual

To ease notation, let me use B as (sum_j Beta_ij.x_ij). Then the constraint (1) is B-t <=0... You multiply this with lambda_i (let me use L for short of sum_i Lambda_i) and carry to objective ...
Evren Guney's user avatar
1 vote

Finding lower bound (maximization problem) in Lagrangian Relaxation with subgradient method

The relaxed constraints will typically be violated, but the rest of the constraints will be satisfied. To get a feasible solution to the original problem, you need to modify the relaxed solution ...
RobPratt's user avatar
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1 vote

Combination of lagrangian relaxation and column generation

In theory, nothing forbids it. I don't remember seeing any such example in the scientific literature. But here is an example of an algorithm using nested column generation, i.e. the pricing subproblem ...
fontanf's user avatar
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1 vote

Is there any academic reference which suggests/uses dual values as initialization of Lagrangian multipliers?

Short answer: Here is a paper that uses the dual values Akbari, V., & Salman, F. S. (2017). Multi-vehicle prize collecting arc routing for connectivity problem. Computers & Operations Research,...
SHUVABRATA CHAKRABORTY's user avatar

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