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
48 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
76 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 ...
3
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1answer
101 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 ...
24
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15answers
2k views

Recommended books/materials for practical applications of Operations Research in industry

I have a Masters' degree in Mathematics. I've very fair understanding of methods and techniques of Operations Research. I am looking for a good book/material where I can see a lot of examples on Math ...
-1
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0answers
57 views

Any Online Optimization course on Pyomo? [closed]

Has anybody used the Udemy Pyomo Course? It says that it provides examples on Linear programming Mixed Integer Programming Non-linear Programming Multi-Objective Programming https://tinyurl.com/...
4
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1answer
35 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 ...
3
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2answers
651 views

Fast algorithm for Transportation Problem in Python?

The Transportation Problem can be solved with a simplex algorithm, but it's time-consuming. I'm wondering if there exists a specific Python-implemented algorithm with low complexity.
2
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0answers
44 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: ...
3
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1answer
170 views

How to optimize a utility function that contains step function?

I have an optimization problem with an uncommon utility: to find a $\beta$ that maximizes $$ r^{T}\cdot H(X\cdot\beta) $$ where $H()$ is a Heaviside step function as in wiki $r$ is a vector of size ...
16
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1answer
548 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 ...
3
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1answer
83 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(...
6
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3answers
284 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, ...
6
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1answer
74 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 ...
2
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0answers
85 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 ...
12
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2answers
1k views

Linear programming: objective function with “buckets”

I had a linear programming problem with the following objective function $$f(x) = \sum_{j}x_jq_jp_j - \sum_{i}\left(\sum_{j}x_jq_jC_{ij} \right) c_i$$ Where $q, p, C, c$ are known. This problem was ...
9
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2answers
357 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 "...
3
votes
0answers
75 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 ...
15
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2answers
308 views

How can I approximate a chance constraint in a computationally tractable way?

I want to solve an optimization model that contains a constraint like $$ \Pr[F(x,\xi)\leq0]\geq1-\varepsilon $$ where $x$ are my decision variables, $\xi$ is a random vector, and $\varepsilon\in(0,1)$ ...
4
votes
0answers
60 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+\...
2
votes
1answer
72 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 ...
3
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0answers
56 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 ...
8
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1answer
140 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 $\...
4
votes
2answers
87 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 $...
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1answer
66 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
291 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
55 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,...
5
votes
1answer
110 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: $$\...
6
votes
1answer
186 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.) ...
2
votes
2answers
58 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
votes
1answer
37 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
votes
3answers
330 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 ...
0
votes
0answers
74 views

Linearization of constraints with square root

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 ...
5
votes
0answers
219 views

Substituting inequality by equality constraints

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 or column has all its entries equal to zero. Let also $\mathbf{k}=\left(k_j\right)$ be a ...
1
vote
1answer
619 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 ...
2
votes
1answer
101 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, ...
1
vote
1answer
81 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\\\...
4
votes
1answer
83 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 ...
6
votes
3answers
593 views

How to determine if this problem is NP-HARD or NP-COMPLETE?

Suppose that I have a pool with N nodes and I have to move the nodes one by one to another pool. For each move, consider a value on the edge linking the two pools. The goal is to find a order of nodes ...
6
votes
2answers
405 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
votes
1answer
87 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 ...
3
votes
0answers
25 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
54 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 ...
1
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0answers
25 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 ...
13
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1answer
291 views

Categorization of optimization models

For many families of optimization problems there is some sort of classification scheme. I am thinking about the triple notation for machine scheduling introduced in "Optimization and approximation in ...
2
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0answers
53 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 ...
4
votes
3answers
683 views

Branch and bound algorithm programming code

I want to solve an integer programming problem using the branch and bound method, but I'm having trouble finding the programming code. From what I saw, almost all algorithms use it for traveling ...
0
votes
2answers
194 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
votes
1answer
83 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 ...
27
votes
9answers
10k views

MATLAB vs. Python in industry

I am a beginning PhD student in math, and I would like to focus on optimization. I am learning programming for the first time, and I have written out some rudimentary optimization algorithms in both ...
3
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1answer
71 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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