Questions tagged [linear-programming]

For questions related to problems that optimize (i.e., minimize or maximize) a linear objective subject to linear constraints.

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47 views

How can I formulate an LP or heuristic solution for this problem?

[I welcome any alternate or simplified formulation of my problem] I have an optimization problem. See the attached figure which is self explanatory. The solid line is intended signal, the dashed lines ...
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0answers
46 views

Reading MPS file for linear programming and reconstructing the Optimization model

Are you aware of any tutorial that can help me learn on how to reconstruct the objective function and constraints from a MPS file once it's loaded in MATLAB. I can load the mps file given to me and ...
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1answer
57 views

Maximum bipartite matching with breakpoints in edge weight function

I am looking for an analogy to the problem I am facing or better yet a paper or even code. I have: Nodes from set A and B. Edges are from a single A to many B. I am framing a max bipartite matching ...
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2answers
162 views

How can I formulate this specific if-then constraint?

IF $\sum\limits_d X_{i,d}\ge6$ THEN $Y_i = 1$ (strictly) AND IF $\sum\limits_d X_{i,d}<6$ THEN $Y_i = 0$ (strictly) $X$ and $Y$ are binary variables. What I'm actually trying to do is to charge the ...
5
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0answers
100 views

Obtaining the system of irredundant inequalities from a set of inequalities using CPLEX

Given a linear system of inequalities $Ax \geq b$, I would ideally like to compute the irredundant set for those set of inequalities. I know how to do so mathematically, but I was wondering if there ...
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5answers
3k views

Is there a Linear Programming Library that natively supports fractions instead of floating point arithmetic?

If one recalls how the Simplex method is taught by hand in most LP classes it takes place entirely in $\mathbb{Q}$. All operations yield exact fractions. For this reason I'm looking for linear ...
3
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0answers
44 views

Simplex method on graphs: How do I find a basic solution using the Ford-Fulkerson algorithm?

I'm tasked with solving a minimal cost flow problem. I'm asked to first use the Ford-Fulkerson algorithm on my graph to find a basic solution that will then be used to do the simplex method on that ...
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0answers
53 views

Estimating multistop routing costs

In many OR problems, it is sometimes a good idea (or necessary) to relax routing constraints. An example of this occurs in the classical facility location problem, where a warehouse can send out a ...
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1answer
154 views

Is optimal solution to dual not unique if optimal solution to the primal is degenerate?

If optimal solution to the primal is degenerate, does it necessarily follow that optimal solution to dual not unique? That is, is uniqueness an unnecessary assumption? Spin-off from here. In my ...
4
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1answer
74 views

How to linearize $f(x,y) = (ax+by)/(x+y)$?

I have a problem which is mainly linear but it has a non-linear component. The objective function is obj = Linear_term + $c*f(x,y)$ where, $f(x,y) = (G_1 x_1 + G_2 x_2)/(x_1 + x_2)$. The decision ...
4
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1answer
65 views

the set of optimal solutions of a linear programming (LP) problem as a mapping of right-hand side

Consider a linear programming (LP) problem \begin{align} M(b) \in \arg\min_x \{ c^\top x : Ax=b, x \ge 0 \}. \end{align} Suppose the LP is feasible and bounded for all values of $b$. We know that $M(...
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1answer
71 views

How can I find the shortest path for all nodes in a graph from a source $s$?

This is the shortest path problem. I've used a model where we can find the shortest path between the source and a specified destination. The idea behind this model is that we assign a flow of 1 for ...
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0answers
115 views

Is this a valid strong polynomial algorithm for deciding LP feasibility?

Let $$A \cdot X + B \preceq 0$$ be a system of linear inequalities with $X \in \mathbb{R}^n$ $A\in \mathbb{R}^{m\times n}$ and $B \in \mathbb{R}^m$ where $m \geq n$. According to Farkas lemma, exactly ...
6
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1answer
231 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 ...
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1answer
92 views

Minimum value for a group of variables in linear programming

I want to use linear programming to assign weights to a number of groups of variables. Let's assume we have group $A$ with $x,y,$ and $z$ and group $B$ with $m,n,$ and $p$. Is it possible to define ...
5
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1answer
58 views

Maximizing a Ratio/Percent

I'm using cvxpy to model a problem. Inside a very large and complex LP, I create two continuous, affine (unconstrained) expressions: $x$ and $y$. Due to how they ...
3
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1answer
92 views

Derive “true” shadow price for degenerated LPs using commercial solvers (e.g. Gurobi)

In linear programming for an optimal primal degenerate solution the values of the dual variables are in general not identical with the corresponding shadow prices. Several proposals on how to find the ...
5
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1answer
236 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 ...
3
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1answer
72 views

Which method to use to solve this multi-objective conflicting objectives

I have the following multiobjective problem. I need to minimize the user-perceived latency while doing so aggressively minimizing user-perceived latency generates large switching cost (Reconfiguration ...
2
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1answer
37 views

Finding bounds on a data sensitivity scenario ILP problem

This is a follow up to a problem I posted here: Modelling a data-sensitivity scenario as an ILP problem As a recap, I was interested in finding the minimum number of cells that need to be suppressed ...
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0answers
84 views

Condition for an integer program and its linear relaxation to have the same value

Let $A$ be a $(0,1)$-matrix where no row or column is a zero vector, and consider the following optimization programs \begin{align}(1):\min&\quad y\cdot1\\\text{s.t.}&\quad yA\ge w\\&\quad ...
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1answer
86 views

Why is the Lagrange Multiplier not equal the Shadow Price (Excel solver, Matlab linprog, Gurobi)?

I have a LP with equality and inequality constraints. When solving the LP with the excel-solver (GRG Nonlinear) the sensitivity report returns the lagrange multiplier for all constraints. When solving ...
3
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2answers
92 views

Modelling a data-sensitivity scenario as an ILP problem

I am new to linear programming, and I recently came across the following exercise, which I do not know how to solve: When publishing data, it is sometimes important to "suppress" sensitive ...
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1answer
72 views

Can CPLEX output LP solution with primal variables only?

CPLEX writeSolution method outputs both primal and dual variable values of a LP. I know how to parse through the solution file to extract specific primal variable ...
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0answers
78 views

Expansion heuristic using gurobi reduced cost / shadow price (LP)

Gurobi 9.0.0 // C++ // LP Let us assume the following problem with three nodes: (1)-----(2)-----(3) node (1) is producing ...
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2answers
80 views

How this problem can be defined as MultiObjective optimisation

I need to optimize the end-to-end latency of a multi-component application. Assuming that the application has 10 components, component 1-5 is hosted by device 1, and device 2 is hosting the other 5 ...
3
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3answers
98 views

How to write distance specific constraint?

Suppose there are a few plants (p) and few customers (c). The supply (Sp), distance (Dpc), cost (COSTpc) and demand (DEMANDc) between them is given. I have a constraint that 90% of total demand of all ...
4
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1answer
71 views

How would you linearize this scheduling problem? Or how would you solve this? It is variation of a set coverage problem for OpenSolver

So, it's been about 15 years since I took my OR class in college. I'm not versed in any programming language besides a little bit of VBA. A client of mine is looking to solve the following problem. I ...
2
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1answer
113 views

Is my formulation correct and how to formulate this IF-THEN constraint?

I have system with $N_U$ users and $N_T$ transmitters. Multiple transmitters can transmit to a single users and one transmitter can transmit to many users, i.e., two sets of transmitters serving two ...
4
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1answer
89 views

Formulating these logical constraint in an ILP

I have these two constraints : $z \leq My$ $t \leq M'y $ where $z$ and $t$ are two integer variables $ z, t\geq 0$, $y$ is a binary variable, and $M$, $M'$ are two big numbers. So basically these ...
2
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1answer
117 views

Linearize sum of continuous and boolean variable

For maximizing the objective function $\sum_i{d_i y_i}+ A x - B \cdot \mathbb{I}_{x>0}$, how can I linearize $ A x - B \cdot \mathbb{I}_{x>0}$ term where $d_i, A$ and $B$ are positive constants ...
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1answer
90 views

How can I have minimum amount of resources wasted in this resource allocation problem?

I have a demand, $d$ I also have supply from 1000 sources. The supplies from those $N$ (for example, $N=1000$) sources are given by $s_1,s_2,s_3,\cdots,s_N$. So,the total supply is : $s_1+s_2+\cdots+...
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0answers
40 views

High-mix manufacturing capacity

I'm not an expert in OR but I would like to determine what is the maximum manufacturing capacity of a plant (or how much a plant can produce of mix products). Each person in the plant has a known set ...
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0answers
74 views

Provide basic solution to CLP

I'm using Pyomo to formulate an LP with approx 500,000 constraints and 200,000 decision variables. The LP is solved using CLP. Some instances fail to return even a feasible solution after many ...
6
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2answers
117 views

Simplex algorithm and extreme points

For this question my short-hand is LP = linear program, BFS = basic feasible solution, SEF = standard equality form. Since the Simplex algorithm iterates from extreme point to extreme point (...
3
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1answer
104 views

Logical constraint in ILP

I want to write the following constraint: Let $z$ be an integer variable such that $0\le z\le M$, and $t$ be a binary variable where $M$ denotes big-M. The logical constraint is as follows: if $z \...
3
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1answer
123 views

How to express this logical constraint for an ILP?

I am trying to write an ILP for a problem but I have this logical constraint and I'm stuck. In my model I have: two binary variables: $x$ and $y$ One Integer variable: $z$ The logical constraint I am ...
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0answers
29 views

Linear functions in Lenstra's algorithm

I had asked this question at MathOverflow and was pointed here. I'm working on implementing Lenstra's algorithm. At the bottom of p.5 (at "construct $n+1$ linear functions"), he says to ...
5
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1answer
52 views

Min-cost flow with per-edge flow conservation

I am trying to solve a linear program that is identical to a min-cost flow problem, except for a difference in the flow-conservation constraint. Instead of the summed outgoing flow equaling the summed ...
4
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2answers
124 views

Faster implementation of “or” constraints in ILP

I have implemented a set of "or" constraints in my ILP using binary decision variables (as in this method). It works fine for smaller problems, but when I try to increase the number of ...
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0answers
111 views

Resource constrained LP problem

I am working on Diet Allocation problem. Where we need to provide food piece for protein deficiency. The data is as below: ...
5
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1answer
227 views

Simplex Multiplier

I am reading through a book which provides an example of a linear program given by \begin{align}\min&\quad-24y_{1}-28y_{2}\\\text{s.t.}&\quad6y_{1}+10y_{2} \leq 2400\\&\quad8y_{1}+5y_{2} \...
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0answers
274 views

Canonical form of a linear program

I have a linear programming problem that I want to write in the canonical form: \begin{align}\min&\quad c^\top X\\\text{s.t.}&\quad A\cdot X\le b\end{align} The problem is given by \begin{...
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3answers
553 views

How to find all vertices of a polyhedron

I have a convex polyhedron given by a set of linear inequalities, for example: $$ x_1 \geq 0,~~ x_2 \geq 0, ~~x_3\geq 0 \\ x_1+x_2\leq 1,~~ x_2+x_3\leq 1,~~ x_3+x_1\leq 1 $$ I want to list all the ...
3
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1answer
75 views

Find a particular optimal solution

After writing an integer linear program in AMPL, I solved it using CPLEX. Now, I have some variables that must necessarily be 1, others that must necessarily be 0 and finally it is possible that some ...
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0answers
39 views

Verifying the correctness of KKT conditions

I have a LP problem and derived the corresponding KKT conditions for the same. I simulated the LP and obtained the primal and dual values and manually checked if the KKT conditions hold. Is there any ...
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0answers
67 views

Finding Optimal Route using different Paths

I have a list of paths e.g. path 1 takes you from point A to B. A person needs to complete 5 of such paths. $$Route1 = path1 + ...
6
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2answers
147 views

Linear objective function with non-linear constraints

I would like to choose a set of $\beta_j$s that maximizes a simple linear objective function of the type $$ \underset{\beta_j}{\operatorname{max}}\sum_{j=1}^{J}X_j\beta_j \\ $$ subject to the ...
5
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2answers
135 views

What is a general procedure to prove that the LP relaxation of an IP delivers the optimal IP solution?

Say that I have a binary IP $$z=\max_x \{c^\top x: Ax=b, x\in B^n\}$$ where $B^n$ is the set of $n$-dimensional $0-1$ vectors. Its LP relaxation will be $$z^{LP}=\max_x \{c^\top x: Ax=b, 0\leq x\leq 1\...
5
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1answer
114 views

Linearizing a constraint with square root of a variable

I am trying to linearize the constraint set (2) in the following simplified program. The parameters: $A,C,D,T\in\mathbb{R}^+$. The set $\mathcal{J}$ is polynomially-sized. \begin{alignat}2\min &\...

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