Questions tagged [mixed-integer-programming]
A mixed-integer programming (MIP) problem is a linear program where some of the decision variables are constrained to take integer values.
453 questions
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Computing Minimum Mean Cycle with Specific Edge
A little context: I am implementing a branch and cut algorithm and I have a separation routine, where I construct a digraph and have to run a minimum mean cycle algorithm to check whether some ...
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How to denote common condition for a group of constraints in MIP?
I am formulating in latex a mixed integer programming (MIP) problem (i.e., defining the objective function, decision variables and constraints).
Among the problem constraints, I have the following set ...
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Column generation and reduced costs
Suppose I have a Master Problem (MP) with several inequality constraints for the decision variables, e.g.
$$\min c^Tx \quad \text{s.t.} \quad Ax \leq b, \quad \Vert x\Vert_1 \leq r, \quad x\geq 0.$$
...
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Mixed integer model predictive control for exploration and exploitation planning
I have a dynamical system $\mathbf{x}_{k+1}=\mathbf{f}(\mathbf{x}_k,\mathbf{u}_k)$ tracking some pre-computed trajectory, $\mathbf{x}_t = (\mathbf{x}_{t,1},\cdots,\mathbf{x}_{t,K})$. Suppose we just ...
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Find all Integer points inside a bounded domain
Given positive integer bounds $m_1,m_2,...m_n$, irrational numbers $p_1,p_2,...,p_n$ and a small real number $R > 0$, find all solution sets $k_1, k_2,..., k_n \in \mathbb{Z}$ satisfying $$0 < ...
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Linearization of Linear Fractional Constraint
Can this constraint be converted to a set of linear constraints:
$$
z_j \leq \sum_{c \in C} \left( \mu_c \cdot x_{cj} + (1 - \mu_c) \cdot \frac{L_{cj} \cdot (1+\beta_{cj})}{\sum_{k \in S} L_{ck} \...
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Multi-objective optimisation methods suitable for LPs and (M)ILPs
Which methods (classic/modern) are utilised to solve multi-objective optimisation problems compatible with linear programming (LP) and mixed-integer linear programming.
Utilised in the context of time ...
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How is it useful in branch-and-bound to focus on lowering the global dual bound?
It seems to me that if you want branch-and-bound to be highly efficient then you should try to determine good solutions (primal bounds) as fast as possible so that you can prune more subtrees on the ...
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Mixed integer programming formulation of point cover with specified miscover rate?
Imagine a set of $N$ points in $R^m$ labeled $y_{+}$ or $y_{-}$. The task is to pick a corner of $m$ dimensional box $(x_1,x_2,...x_m)$ such that rectangle $(-\infty,-\infty,...,-\infty)$ to $(x_1,x_2,...
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0-1 Linear programming and non-optimal multidimensional knapsack
I would like to create a set of constraints forcing a set of knapsacks to be filled. The knapsacks should be filled, so that no further element of a set of elements fits into it. It is not a classical ...
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Reformulating Mixed-integer Bilevel program into MINLP
I am working on a problem where I have this Bilevel programming problem:
$ Max \quad a+b $
$s.t.\quad \quad \alpha \in \{0, 0.5, 0.8\} $
$\quad \quad \quad \; \ a = min \ \lambda$
$ \quad \quad \; \ ...
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Find the optimal solutions to a system of linear equations?
I have a linear optimization problem $\mathbf{A}\cdot \mathbf{x} < \mathbf{0}$, where $\mathbf{A}$ is a particular square matrix for my application, and $\mathbf{x} \geq \mathbf{0}$.
I want to ...
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Formulate weight lifting as MILP problem
Consider two variables $x_1$, $x_2$ describing how high a weight is in two succeeding states. I need to minimize effort of lifting the weight, but I don't care about dropping the weight: $\min w\cdot\...
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Tractable formulation of a mixed integer program
Given constant matrices $A_1\in\mathbb{R}^{1\times l}$ and $A_2\in\mathbb{R}^{1\times l}$, and constants $b_i$, $i=1,\dots,n$. Consider the following mixed integer program (MIP) with decision ...
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Task Assignment Problem using MILP (tasks >> agents)
I have a general assignment problem that assigns a set of payload tasks $T$ to a set of workers $A$, where $|T|$ >> $|A|$. Each task $T_i \in T$ consists of a tuple $(s_i, g_i)$, which represent ...
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Non-linear optimization programming, with step function in constraint
I want to optimize a non-linear function $f(x)$, $f: \mathcal{R}^{n} \to \mathcal{R}$ (being a log-likelihood over $m$ observations, i.e. $i$ being the observation index) under constraints numerically,...
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How to linearize a max function in a constraint? [closed]
I have linear program that has constraint as follows:
$
\max(x,y) \geq 0
$
where $x$ and $y$ are variables.
How to linearize this inequality?
How to write this constraints in google or tools?
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How many hexagons to fill a square tile
I am filling a square tile of width wTile with equal hexagons stacked flat side on top of each other at an angle I call colourAngle as shown in the diagram. I call the rows of hexagons "Perp Line&...
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Mutual exclusivity variables in mixed integer programming problems
I am working on a employee scheduling problems (assigning shifts to temporary workers) by modeling it as a MIP. There is a one shift per day constraint for the employees that restricts more than one ...
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Indicator function with multiple conditions in optimization
I have the following problem
$$\begin{align*}
& \min \ f(X) \newline
& X = \begin{cases}
1&; x_1 \leq c_1, x_2 \leq c_2, x_3 \leq c_3, \newline
0&; \text{otherwise}.
\end{cases} \...
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How to linearize or formulate optimization constraints that are stated in terms of "if-then" sentence?
I am a engineer who is working on an optimization problem and my constraints are in the form of this statement "if $x_1=1$ then $d_1=1T$" where $T$ is just a given time period.
Scenario 1
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The facet-defining inequalities for a single resource scheduling problem
Suppose, there exists a scheduling problem $S$, in this case a single resource, with the following descriptions:
$$ \text{conv(S)} = \{x \in \mathbb{R}^n \ | \ \forall \lambda_{i} \in \mathbb{R}^{n+}, ...
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Maximize sum of absolute values over a box set
I am interested in the following linear problem:
$$
\begin{array}{cl}
\max & |a_{11} x_1 + a_{12} x_2| + |a_{21} x_1 + a_{22} x_2| \\
\mathrm{s.t.} & 0 \leq x_1 \leq b_1 \\
& 0 \leq x_2 \...
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How to apply integer cut to a simple MILP?
I'm self-studying on cutting plane methods, and I'm reviewing the following problem from Bertsimas' book (see below). I know what cutting plane methods do, and how they eliminate infeasible solutions ...
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MILP constraint for connected node selection
I am formulating constraints for a network as shown in Figure .
Blue circles represent a set of nodes, $N = \{1, 2, \ldots, 5\}$. Three different types of devices are connected to different nodes in ...
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