3
votes
Reasonable implementation of a MILP scheduling problem
Your formulation looks correct, but the objective is nonlinear because it includes a $\max$ function. You can linearize by introducing nonnegative variables $z_j$, imposing linear constraints $z_j \ge ...
- 33.6k
3
votes
Accepted
Column generation for bin packing problem with variable size minimizing the largest cost of any class
Yes, you can. Apply Dantzig-Wolfe decomposition to your compact formulation, with the second and third sets of constraints as your subproblem, which then decomposes into one block per bin $b$.
- 33.6k
3
votes
Is there any way to use Lazy constraints in Pyomo?
Regarding HiGHS, it does not support lazy constraints at the moment, but this is work in progress. They will be available in one of the next releases. Once they are available, you will be able to use ...
- 76
3
votes
Accepted
Implication linking triplets of non-negative real or integer variables
Part i) was already answered here: https://or.stackexchange.com/a/12403/500
For part ii), let $\bar{x}$, $\bar{y}$, and $\bar{z}$ be constant upper bounds on $x,y,z$, respectively. Let $\epsilon$ be ...
- 33.6k
3
votes
Deviation from the optimal solution for Solomon instances of CVRPTW
Here is how you should compute the distance d between node i and node j:
...
- 2,673
2
votes
Accepted
Relation linking pairs of non-negative continuous or integer variables
Let $\bar{x}$ and $\bar{y}$ be constant upper bounds on $x$ and $y$, respectively.
Let $\epsilon$ be a small positive constant. Introduce binary variables $z_i$, and impose
\begin{align}
z_1 + z_2 + ...
- 33.6k
2
votes
Accepted
Are there ways to assist Gurobi in finding good solutions?
You can use a callback to get each new Gurobi solution $x$ and tell Gurobi about your improved solution $g(x)$. This article illustrates it in the context of the Python API. (It might be a bit, um, ...
- 40.1k
2
votes
Accepted
Modeling the relation between pairs of Non-negative Continuous or Integer Variables
Assume $x \le M$ and $y \le M$, and let $\epsilon>0$ be a small constant tolerance.
For i), introduce binary decision variables $u,v,w$, and impose
\begin{align}
x &\le M u \tag1\label1 \\
y &...
- 33.6k
2
votes
Accepted
Casting the truth value of an inequality to a boolean
Equivalently, you want to enforce the contrapositives
\begin{align}
b=0 &\implies d<0 \\
b=1 &\implies d\ge 0
\end{align}
For $d<0$, rewrite as $d\le -\epsilon$ for some positive ...
- 33.6k
1
vote
How to Implement a Piecewise Function Using SOS2 in GAMS Without Binary Variables in an MILP Model?
This is not a piecewise linear function, as expected by a SOS2 approach. SOS2 assumes that breakpoints are fixed and that there is some interpolation to do. You have a piecewise constant function ...
- 2,850
1
vote
How to Implement a Piecewise Function Using SOS2 in GAMS Without Binary Variables in an MILP Model?
If your $h0()$ function only takes two possible values (0.001 or 0.3), an SOS2 constraint would be inappropriate. You would need to use a binary variable. In any case, a solver that allows SOS ...
- 40.1k
1
vote
Are there ways to assist Gurobi in finding good solutions?
I would suggest watching Faster MIPs Using Custom Heuristics by Gurobi's Greg Glockner.
- 175
1
vote
Accepted
Infeasibility of subproblem in Combinatorial Benders Decomposition
If imposing $u^T \bar{b}=1$ makes the dual problem infeasible, it means that $u=0$ is the only feasible solution to the original dual and there is no certificate of infeasibility for the primal. That ...
- 33.6k
1
vote
Accepted
Modeling the implication between pairs of Non-negative Continuous or Integer Variables
The first disjunct, $(x \leq y \implies a \leq b)$, would be linearized as the following form:
\begin{align*}
\ x-y \leq M(1-z_1) \\
\ a-b \leq M(1-z_2) \\
\ z_1 + z_2 = 1 \\
\ Lb \leq x, y, a, b \leq ...
- 9,248
1
vote
Reference for column generation applications
Another recently published reference by Eduardo Uchoa, Artur Pessoa, and Lorenza Moreno is Optimizing with Column Generation.
We are excited to present the early release of Part I of our book
“...
- 9,248
Only top scored, non community-wiki answers of a minimum length are eligible