# Questions tagged [mixed-integer-programming]

For questions about mathematical optimization problems involving both continuous and binary or general integer variables.

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### No, Gurobi, I really do want this variable to be binary

When I mark a variable in a Gurobi MIP model as binary, sometimes Gurobi gives me a solution where that variable has a fractional value other than 0 or 1. How do I constraint a variable to be honest-...
166 views

### Strong MIP formulations for a large-scale mixed-integer nonlinear feasibility problem

I'm trying to construct a strong MIP formulation for the following integer nonlinear feasibility problem. Informally: We have a $m \times n$ decision matrix of binary variables Each row of the matrix ...
163 views

### Why is there a constant in the objective function of the *best subset selection problem*?

This article presents the following formulation of the best subset selection problem $$\min_{\|\beta\|_0\leq k}\frac{1}{2}\|y-X\beta\|^2_2$$ I wonder where the $1/2$ comes from. Help appreciated.
71 views

### Expressing a chain of boolean if-then with logical ANDs using MIP

How to express a chain of boolean If-then as MIP such as: If $(x_{10} \ge b_1$ and $x_{11} \le b_1)$ AND $(x_{20} \ge b_2$ and $x_{21} \le b_2)$... AND... then $y_1 = 1$ else $y_1 = 0$. So basically,...
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### No solution found from n MIP starts

Currently, I am working on the implementation of a MILP formulation with lazy constraints for an optimization problem, and I am using the MIP Start strategy. I am generating the integer feasible ...
226 views

### Interface for Cbc - COIN-OR

I would like to code some IP/MIP models in python and test them with an open-source solver. As of now, I only know the Cbc - COIN-OR open-source solver. I have already tried the or-tools interface, ...
158 views

### How to treat a system of bilinear constraints

A model contains constraints of the following form $R(k) \leq X(k) G(k)$ where $X(k)$ binary and $G(k)$, $R(k)$ non-negative variables. The index $k$ runs from $1$ to $50$. I linearise the equations ...
139 views

### Optimising the current model

After developing the MIP model I noticed that solver is taking a lot of time to reach the solution. So, how should I approach to optimize the current model? Are there any visualization tools or any ...
71 views

### If $t\le0$ then $P=1$, if $t > 0$ then $P =0$ or $P=1$

I am trying to model $t \leq 0.0 \implies P = 1.0$ else $P=1$ or $P=0$ where $0 \leq t \leq H$ is a bounded nonnegative real, and $P$ is binary. I can use the expression $t + \epsilon P \ge \epsilon$ ...
155 views

### How to propagate time using linear inequalities?

I have an adjacency matrix $G_{i,j}$ that tells the distance between $i$ to $j$ (between 0 to 1) if there is no edge between $i$ to $j$ I am putting a large integer $100$. This is my previous ...
237 views

### A heuristic approach to solve a MILP problem?

I have the following optimization problem which is a MILP. I can solve it with a MILP solver. This one I posted here Is there a heuristic approach to the MILP problem? Since I have an additional but ...
101 views

### Upper and lower bounds of a variable equal

I'm working on a MILP (Mixed-Integer Linear Programming) problem with the Java API of Cplex. In order to easily exclude some variables from my problem I thought about setting both their lower and ...
181 views

### Guides for strong MILP Formulations

When developing MILPs, often there are different alternatives possible to express a constraint. The question then arises, which of the alternatives is better, meaning, which alternative is expected to ...
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### Are there strategies/rules/systematic approaches to derive alternative formulations for an optimization problem?

For a given optimization problem there are often many known math programming formulations. For example for the TSP here is a survey https://link.springer.com/chapter/10.1007/3-540-36626-1_5 of many ...
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### What is the definition for an integer-friendly constraint?

I have seen some papers claiming that their proposed model is integer-friendly. I would like to get more information about what type of constraints we can call integer-friendly. Probably, it can be ...
383 views

### Creating a Decision Variable in Python-MIP with Dictionaries

I have already seen in the Python-MIP documentation on how to implement a variable with multiple indices. In each example I read, this is done with the use of lists and integers as indices. This is ...
247 views

### Multi-period linear dynamic programming with differing in-period dependencies and changes

I’m not sure if I’m wording this right but in a nutshell, my problem is: I’m modelling potential actions a boat owner can do to their boat. Let’s say he wants to know over the 50 year lifespan of the ...
286 views

### Can GLPK be used to solve an optimal team selection problem?

My Problem I am quite new to optimisation, so any advice is appreciated. I am currently trying to solve a problem as follows: Given a pool of people, we want to create n teams such to find the optimal ...
195 views

### Does the weighted sum approach find all pareto-optimal solutions in MILP

I use the weighted sum approach for a multiobjective optimization problem that is formulated as a MILP. This means that the objective function is linear. I read quite often that the weighted sum ...
214 views

### Convert summation of min functions into linear constraints for optimization

I have the following optimization problem:  \mbox{maximize } j^{*} \mbox{ subject to:} \sum_{j^{*}\leq j\leq J} \min({\bf A}_j,{\bf B}_j) \geq \lambda, \lambda \in \mathbb{R} \mbox{ and } {\bf A}_j,{...
279 views

### Advanced Heuristics for MIP vehicle routing problem in PuLP

I am currently trying to speed up an MIP. An approach I was considering was to implement a cut callback heuristic with PuLP (one which rounds relaxed integer variables greater than .9 to 1). ...
154 views

### Dual of a model to obtain reduced costs

I have the following model which I am going to solve with column generation. Basically it is a VRP problem with pick-up and delivery in which couriers are assigned to a set of shifts based on their ...
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### Water quality component optimization

I have an optimization problem that I'm attempting to tackle. As you can see in the image below, there's a graph network through which water flows. I've drawn out the problem in the image to explain ...
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### What are useful plots/statistics/metrics when analyzing the solution sensitivity in multi-objective optimization?

Consider an optimization problem with $n>3$ objectives. For handling this there exists often two approaches: a) some weighting of the objectives, b) fix an order of objectives and then optimize ...
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