Questions tagged [optimization]

For questions involving mathematical problems that aim to minimize or maximize some objective function, possibly subject to one or more constraints.

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1answer
436 views

How to learn Optimization?

I am a 2nd year PhD student in a (mostly) pure Mathematics department. I do not have any prior experience in applied mathematics, but I've recently had a change of heart and decided to study ...
2
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1answer
86 views

Converting evolutionary optimization problem from Excel to Python

I've set up and successfully executed an evolutionary optimization in Excel, and now have a need to convert the problem to Python. differential_evolution() in the ...
4
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1answer
69 views

How to convert static variables into arrays for use with PuLP

I have the following code in Python and PuLP which uses static variables. I want to know how to solve the problem by converting all of the LpVariable parts into an array, as well as the constraints. ...
2
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1answer
48 views

Logical equivalencies to modeling an indicator decision variable in transportation problem

I am formulating a model that seeks to minimize the cost of shipping goods from factories to warehouses, where the cost of shipping is independent of the type or amount of goods being shipped (except ...
0
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1answer
46 views

What is a good approach to deciding which jobs (from a list of HPC jobs) should be ran locally vs. on the cloud given time & cost constraints?

Cloud computing has transformed the landscape of compute operations. Of course, there are still many labs/businesses with local, large-scale compute clusters. For those businesses who keep the ...
3
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0answers
51 views

Where I can study some job shop scheduling by course (video )?

I am seeking the help to know where I can study the job shop scheduling Heuristics or using solver by some course/video as I see some of books and papers hard to understand . It is hoped that the ...
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0answers
63 views

How to start the Dantzig-Wolfe decomposition?

I have the following problem: \begin{align}\min&\quad3x_1+5x_2+3x_3-2x_4+3x_5\\\text{s.t.}&\quad x_1+x_2+x_3+x_4\geq3\\&\quad3x_1+x_2+5x_3+x_4-2x_5\geq6\\&\quad x_1+2x_3-x_4\geq2\\&...
2
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1answer
79 views

Formulating “more complicated” objectives in Python Gurobi

I am currently learning how to use Gurobi for Python using their official tutorial found here. In this example, they appear to formulate the objective by simply specifying a cost dictionary to the <...
0
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1answer
43 views

Adding slack nodes to min cost network flows

I have the following question. I want to clarify couple of points. As you can see, total demand and total supply does not match, we do not have enough demand. What I want to ask is: Do we need to ...
2
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0answers
72 views

Solving transportation problem by the Network Simplex

I am trying to solve the following problem using Network Simplex method. But I have questions. My attempt: Basis Matrix$(B)$ Rows: 1, 2, 3, 4, 5 Column: (1,3) (1,4) (1,5) (2,3) (2,4) (2,5) $$ \...
2
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2answers
241 views

Hard to soft constraint

What are all necessary changes to the below model so that constraint set $(2)$ becomes soft? Give the new full model. $\alpha_{ij}$ are parameters. Eqn. $(1)$ refers to an already existing function of ...
1
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1answer
61 views

KKT for second order approximation of $f(x)$

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}.$ Consider second order approximation $f(x) \approx f_0(x)$ where $$f_0(x) = f(x_0) + \nabla f(x_0)^T (x-x_0) + (\mathrm{H}f(x_0)(x - x_0))^T(x - x_0)$$ ...
5
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2answers
89 views

Solve nonlinear programming problems practically

In an exam, I studied Theoretical approaches to converting constrained minimum problems into unconstrained minimum problems. Specifically: KKT conditions Projection Gradient Descent Penalty and ...
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0answers
52 views

CPU time on Linux with Gurobi

I am solving a MILP model in C++ using Gurobi 10.1. I retrieve the CPU time of my C++ program under Linux via the following commands: ...
4
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1answer
51 views

Greedy algorithms for assignment problems — prediction doesn't match simulation

I'm considering the following basic assignment problem: a group of $n$ people is to be assigned, in one-to-one fashion, a set of $n$ jobs. Write $C_{ij}$ for the cost incurred when person $i$ gets ...
16
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1answer
591 views

Deep Reinforcement Learning for General Purpose Optimization

Recently, I attended a very nice talk given by someone at the place I work about applying Deep Reinforcement Learning (DRL) for a design optimization problem. It was particularly interesting to me ...
4
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1answer
99 views

Linear objective function with power term in constraint

Given $n$ variables $x_{i}$ where $i\in [0,n)$, denoted as a vector $x$, given a linear objective function that we want to minimize $c^\top x$ with 2 constraints: $\sum x_{i}^{2} < n+1$ $\sum\log(...
5
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1answer
82 views

Prove that $x^*$ is an optimal solution where $f_0$ is $C^1$ and convex and $f_i$ are $C^1$ and strictly convex functions

Let $x^*$ be a feasible solution of the following convex optimization problem \begin{align}\min&\quad f_0(x)\\\text{s.t.}&\quad f_i(x)\leq0,i=1,\ldots,m\end{align} where $f_0$ is $C^1$ and ...
9
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2answers
407 views

How to maximize “contrast” between nodes on a graph?

I have an undirected graph such as the one shown below. I can make up to 3 choices about the color of each node. The edge weights are equal to the difference between the nodes, given by the "...
5
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3answers
386 views

Gurobi and CPLEX cannot exploit more than 32 cores of machine

I have some attempts to solve a scheduling problem using the Gurobi and doCPLEX API in python and .NET on Ubuntu-server installed on a hyper-computing cluster with 64 physical cores. Unfortunately, ...
2
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0answers
128 views

modify pickup and delivery problem to only delivery scenario in google OR-tools

I am using Google OR Tools for solving the capacitated vehicle routing problem.I want to implement optimization solution for delivery problem.For example there is one storage location which load all ...
3
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0answers
64 views

“Rank 1” type constraint $X=vw^\top$: MILP representation? Convex relaxation? Other tractable approach?

Suppose $X\in\mathbb{R}^{m\times n}$, $v\in\mathbb{R}^m$, $w\in\mathbb{R}^n$ are variables from an optimization problem, which also includes the constraints: $$0\le v\le a$$ $$0\le w\le 1$$ $$w_1+\...
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1answer
76 views

How can I formulate an objective function that minimises the number of items required to solve a problem

I am currently trying to solve a problem where I need to minimise transport cost through the choice of vehicle (and how many of each choice) subject to a given demand. The problem: There are currently ...
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0answers
91 views

Prove Non-Homogeneous Farkas' Lemma

Let $A\in\mathbb{R}^{m \times n}, c\in\mathbb{R}^{n}, b\in\mathbb{R}^{m}, d\in\mathbb{R}$. Suppose that there exists $y\geq0$ such that $A^Ty=c$. Question: prove that exactly one of the following is ...
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0answers
61 views

Dynamic Programming problem of affecting equipment with budget constraint

I have a problem that I must formulate as a DP problem and solve. A hospital is split up into 4 sections, each section has 1 or 2 or 3 backup generators. We have to maximize the likelihood that no ...
3
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2answers
94 views

Let $A\in\mathbb{R}^{m\times n},c\in\mathbb{R}^n$. Show that exactly one of the following two systems is feasible:

Let $A\in\mathbb{R}^{m\times n},c\in\mathbb{R}^n$. Show that exactly one of the following two systems is feasible: $Ax\geq0,x\geq0,c^Tx>0$ $A^Ty\geq c,y\leq0$ Assume that A is feasible meaning $...
-4
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1answer
67 views

How to change a function from Min(F(x)) to -Max(-F(x))?

I have not a good knowledge in math field, I am working on multi objective functions, and I have two maximization functions, and one minimize function, where: Max (X,Y) = X+Y Max (L,M) = Sum (LC + MD)...
3
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1answer
319 views

Simplex (GLPK) doesn't find a feasible solution on this simple assignment problem, but there is an obvious one

Problem Assign 11 projects to 11 students, based on their preference. For this example, each students chooses only one project, for simplicity shake (as shown below). Student 1 one chooses project 1, ...
2
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0answers
57 views

Node ordering in Graph optimization

I'm solving a network optimization problem which is modeled as a graph $G=(V,E)$. Solving this problem using Pulp and NetworkX in Python and ordering the graph's nodes in a certain order (i.e. (1,2,3,...
6
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1answer
193 views

Extreme point and extreme ray of a network flow problem

"It is a well-known result in network flow theory that an extreme point and an extreme ray of the polyhedron defined by the convex hull of feasible region corresponds to a path and cycle (resp.) ...
5
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1answer
120 views

0-1 knapsack with non-linear objective function

There's efficient algorithms for solving the 0-1 knapsack problems when the objective function is just a sum of profits. I am dealing with the following problem with non-linear objective function: $$\...
2
votes
2answers
62 views

Parallel scheduling with precedence constraints and variable job length

I have $N$ jobs and $M$ machines and want to minimize the makespan, i.e. the total time to finish all jobs. Some jobs have precedence constraints and can only be started once other jobs are finished. ...
2
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1answer
38 views

Question about theorems of the form “Any limit point of the sequence is a minimum of the problem”

In Optimization, we have problems of the form \begin{align*} &\min f(x)\\ &\text{s.t. }\hspace{0.2cm} x \in S \end{align*} and many theorems are of the form "If the problem satisfies ...
2
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3answers
333 views

Find the farthest point in hypercube to an exterior point

Let $\mathcal{U} = \{ [x_1, ..., x_n] \in \mathbb{R}^n : 0 \leq x_i \leq 1\}$ be the unit hypercube and $C \in \mathbb{R}^n\setminus\mathcal{U}$ fixed. Let us consider the following problem $$ \max_{X ...
1
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0answers
79 views

Linearization of constraints with square root [closed]

I am trying to solve an optimization programming model involving a non-linear constraint with a square root. It follows (in a simplified form): $X_i\ge\sqrt{A_i/B_i}$ where $X_i,A_i,B_i$ are positive ...
2
votes
1answer
107 views

How can I perform discrete optimisation of a variable over a data set

This question relates directly to a dataset I've generated for Fantasy Premier League, but I'm also curious how I can apply this to a more general case. Data I have a list of premier league players, ...
3
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1answer
85 views

Bioinformatics / Genomics Optimization Problems?

I am a third year bioinformatics student and would like to apply my knowledge from an introductory course in Optimization Methods to some problems in the field of genomics or bioinformatics. Do you ...
1
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1answer
84 views

How to prove this convex-optimization problem?

I am struggling with the following optimization problems. Problem 1 \begin{align}\max_{\alpha, s_1, s_2}&\quad s_1 + s_2 - \gamma (s_1 (K_1 +c_1 + s_1) + s_2 (K_2+ c_2 + s_2) + 2\alpha K) +C\\\...
1
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1answer
627 views

How to describe the traveling salesman problem with an integer programming model?

I'm trying to describe the travelling salesman problem as an integer programming model. I'm interested in the asymmetric version of the problem. The problem can be summarized as finding the optimal ...
7
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1answer
147 views

Is the solution of a convex combination of the objective in simple problems a convex combination of the solutions of the same problems?

Let $\mathbf{A}=\left(a_{ij}\right)$ be a $n\times J$ matrix with $a_{ij}\geq 0$, $n>J$ and such that no row has all its entries equal to zero, and each column has at most one zero. Let also $\...
3
votes
1answer
94 views

What are the flow based formulations?

What are the flow-based formulations? For what optimization problems are they applied, and in which form? Which are the specificities of such a formulation? Also, the same question for the time staged ...
6
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2answers
411 views

How to model bicycle sharing scheme?

One of the problems I have recently considered is the problem of rebalancing bicycle stations for bike-sharing schemes all over the world. It is not a secret that the demand for bikes across the city ...
3
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0answers
33 views

R ompr MILPModel array multiplication?

In R, I regularly ompr::MILPModel for optimization. I adapt the below snippet to enable multiplication of a decision variable with two dimensions (e.g., x[i,j] ) ...
3
votes
1answer
56 views

What is the difference between min- cut formulation and (bi) partitioning formulation?

I have a min-cut formulation and a bi-partitioning problem. The two problems focus on finding the minimal cut value separating the two partitions? So what are really the differences between the ...
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0answers
26 views

MILP formualtion for Two-level minimum dominating set (MDS) problem?

I'm working on an optimization problem which is kind of finding the minimum dominating set (MDS) or the minimum vertex set (MVS) in an undirected graph. given the MILP formulation for both problems, I ...
2
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0answers
61 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 ...
5
votes
1answer
84 views

Is the order of edges in graph is changing the optimization result?

I am solving an optimization problem using Pulp and NetworkX. The problem is similar to the Minimum Vertex Set (MVS) problem. I have noticed that the optimizer is Scanning the edges according to their ...
0
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2answers
204 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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2answers
102 views

how to determine differents gap rate?

I found in the literature different gaps: a gap between a random solution and an exact solution a gap between the exact solution and a lower bound a gap between the exact solution and a lower bound a ...
3
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1answer
73 views

Sum of links neighbors in a graph

I'm trying to model this constraint in an optimization problem using Pulp and NetworkX. Here is a piece of code I'm using. ...

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