Optimization with a linear objective function, subject to linear equality and linear inequality constraints.

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What's the complexity of solving a packing LP?

As we know, we can solve general linear programs in weakly polynomial time and it remains open if it is possible to solve them in strongly polynomial time as well. But what is the situation in the ...
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1answer
50 views

What is wrong with my LP exercise (longest path cost for a graph)

I have to do a linear programming exercise but i have some problems regarding the result. I have a graph with N nodes and E edges, that is not acyclic, and each edge is associated to a cost. I have ...
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1answer
23 views

Efficient formulation for binary integer linear programming

Problem: There are two types of balls, big (B) and small (S), which need to packed into boxes. One box can contain either: nothing, or 1 S, or 1 B, or 2 S, or 2 B, or 1 B and 2 S We are given the ...
5
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0answers
60 views

Formulating shortest path as submodular minimization

I'm curious about the general question of whether any combinatorial optimization problem with polynomial time solution can necessarily be reformulated as minimizing a submodular function. The answer ...
2
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1answer
29 views

Why cant we round results of linear programming to get integer programming?

Say if linear programming suggests that we need 2.5 trucks to deliver goods why cant we round up and say 3 trucks are needed. Similarly, if linear programming suggests we can afford only 3.7 workers ...
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0answers
68 views

Assign $m$ tasks to $n$ workers, with $m \geq n$

There are $n$ students that share the same apartment. At each evening, one of them must prepare dinner for everyone. There are $m$ evenings to schedule, with $m \geq n$, and you have to assign any ...
3
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1answer
33 views

Complexity of solving LP with a non-linear growth in variables/constraints

It has been shown that any Linear Program (LP) can be solved in a polynomial number of steps. An example of such algorithm is the ellipsoid method. To solve a problem which has $k$ variables and ...
2
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1answer
31 views

Find value of b

The following system of restrictions is given: $$y_1+ 2 y_2 \leq 4 \\ 2y_1+y_2 \leq 2 \\ y_1+b y_2 \leq 3 \\ y_1, y_2 \geq 0$$ For which values of b is there a degenarate basic feasible solution? ...
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1answer
35 views

Binary Integer Programming question - what graph problem is represented

I'm dealing with a BIP question, that represents a graph problem. The goal is finding the graph problem. I've spend a lot of time trying to solving this question but I couldn't find the answer to ...
4
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0answers
77 views

Can the solution to a POMDP be found using linear programming?

It is known that Markov decision processes (MDPs) can be solved using linear programming (see page 24 of Carlos Guestrin's PhD dissertation). The linear program is: $$min_{V(x)} \sum_x ...
7
votes
3answers
106 views

Cast to boolean, for integer linear programming

I want to express the following constraint, in an integer linear program: $$y = \begin{cases} 0 &\text{if } x=0\\ 1 &\text{if } x\ne 0. \end{cases}$$ I already have the integer variables ...
2
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1answer
69 views

Modeling $(x > 0 \wedge y > 0) \Leftrightarrow z > 0$ in a linear program: impossible?

In this question, we see how to model boolean logic in $0 - 1$ ILPs. Moving to a relaxation, modelling $(x > 0 \vee y > 0) \Leftrightarrow z > 0$ with $x,y,z \in [0,1]$ with linear ...
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1answer
39 views

Are some Integer programming formulations completely useless for relaxation?

I was tasked with constructing an integer programming formulation for an NP-hard problem, and then with specifying its LP relaxation and the resulting approximation factor. The problem is that, while ...
4
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3answers
140 views

Help wrapping my head around a combinatorial optimization problem

Here's the problem I'm trying to solve: I have a bunch of widgets, whose weights vary over a small range. I would like to find the optimal grouping of them such that each group meets a minimum weight ...
0
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0answers
22 views

Mutipath selection in communication network

We have this assignment to be formulated then coded using Integer Linear Programming that is giving me such a headache. The problem is that I'm not sure on how to formulate it because of the way ...
4
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1answer
49 views

Unfeasible linear program becomes feasible if a variable is removed

Apologies, not a computer scientist by trade but I'm playing with linear programming these days. Let $\{x_i\}$ be $N$ optimization variables with bounds $$l_i \leq x_i \leq u_i$$ I'm interested in ...
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1answer
30 views

Branch and bound stanford slides doubt

On the 6th slide at https://web.stanford.edu/class/ee364b/lectures/bb_slides.pdf, while defining L2 and U2, why are we taking min for both?
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1answer
34 views

Restrictions that set binary variable to 1 when integer variable equals x, 0 otherwise

I have this problem: I'm building an integer linear program, which I'm going to give to an ILP solver. I have a binary variable Y which can be either 1 or 0 and an integer variable MONTH which takes ...
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0answers
30 views

Facility location on a Sphere with great circle distance

I am looking for an algorithm to find the point that minimizes the sum of the great circle distances to a set of fixed points on a sphere. In more detail: Given $x_1,\ldots, x_k$ fixed points in the ...
0
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1answer
64 views

Please explain linear programming as seen for this load balancing problem

I have at hand a linear program related to load balancing. However I have no idea what this is and wikipedia gives me something too simplified/abstract to be useful. I wonder if someone can explain ...
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0answers
39 views

Multidimensional 0-1 knapsack as the solution to 0-1 goal programming problem

I am trying to find the algorithm for the 0-1 goal programming problem. Actually I don't have any recent references for explicit algorithms, all the recent articles are about the modelling and not ...
4
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2answers
323 views

Linear programming with absolute values

I know that sometimes we can use absolute values into the objective functions or constraints. Is it always possible to use them, anywhere ? Example of use of absolute values: ...
5
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0answers
58 views

Uniform Sampling on Intersection of Faces of Simplices

I'm trying to sample uniformly on the intersections of faces of several simplicies, with all coordinates being non-negative. That is, given constraints $$A\vec{w}=\vec{b} \ \ and \ \ \vec{w} \geq ...
2
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1answer
123 views

Inventory Routing - Subtour Elimination [closed]

I'm trying to implement a Inventory Routing Problem with Branch-and-Cut. But I'm facing with an issue regarding subtour elimination. (http://www.danflash.com/files/irp.pdf) The paper describes the ...
2
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1answer
66 views

Time complexity of linear program? [closed]

I have built a heuristic algorithm for approximately solving an NP complete graph problem by recursive linear relaxations. In each recursion, the algorithm returns a reduced graph, with number of ...
2
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1answer
52 views

A little confusion with the Knapsack problem (a worked example)

I'm going through a worked example on the Knapsack problem: My problem is that I don't understand quite follow the last bulletpoint. Where does the $x_4 = 4/5$ comes from? I know $x_4$ has to be a ...
5
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0answers
37 views

Authors of Complementary Slackness

Who were the first researchers to prove the Complementary Slackness condition for linear programming? I believe that strong optimality was proved by Gale, Kuhn, and Tucker in 1951, but I couldn't ...
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0answers
31 views

Integer linear program for Ising ground state problem

I am trying to model the Ising spin state problem into Integer linear program and find the optimal ground state using lp_solve. (This is just a miniature version of Ising state problem) $$ maximise: ...
0
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1answer
109 views

Express XOR with multiple inputs in zero-one integer linear programming (ILP)

In the below post, it is explained how to express xor of two variables as linear inequalities. Express boolean logic operations in zero-one integer linear programming (ILP) Naturally, the xor of ...
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1answer
80 views

Can you generate random linear programming problems?

I looked at Linear Programming, and it are problems like this: You know that Cabinet X costs 10 cents per unit, requires 6 square feet of floor space, and holds 8 cubic feet of files. Cabinet ...
2
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1answer
57 views

Does there exist a problem that is hard to do in parallel? [closed]

I am looking for a workload which is hard to paralellise/distribute between multiple machines. For example, integer factorization does not go 10 times faster if you have 10 machines to split the ...
3
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0answers
19 views

What is the most efficient algorithm for finding the bounding inequalities of a cone given the extremal rays?

Say that I am given a cone, as specified by the extremal rays whose facets form its convex hull, what is the most efficient algorithm that finds the linear-rank inequalities whose intersection defines ...
0
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1answer
47 views

Basic linear programming question

Trying to answer this question: A factory produces chocolate and candy. In order to produce 100 kilograms of chocolate, the factory has to use machine A for 1 hour, machine B for 4 hours, and machine ...
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1answer
95 views

Scheduling Problem Optimisation using LP solvers

I am currently working on a scheduling system which schedules individuals based on timeslots that have the following attributes: (1) Day of Week (2) Month (3) Session(AM/PM). The individuals ...
2
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0answers
28 views

How are basic feasible solutions in linear programming related to vertices in its corresponding polytope?

In Section 2.3.3 "Polytopes and LP" of the book "Combinatorial Optimization: Algorithms and Complexity" by Christos H. Papadimitriou, Theorem 2.4 establishes the relation between bfs's (basic feasible ...
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1answer
115 views

ILP and number of variables in constraints

This question is about the time impact of the length (i.e. number of variables) of the constraints in an Integer Linear Programming formulation. Most people try to reach the minimum number of ...
3
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1answer
205 views

Linear programming formulation of cheapest k-edge path between two nodes

Given a directed graph $G = (V,E)$ with positive edge weights, find the minimum cost path between $s$ and $t$ that traverses exactly $k$ edges. Here is my attempt using a flow network: \begin{align} ...
3
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1answer
92 views

Given an optimal solution to the LP, show how it can be used to construct a directed cycle with minimal directed cycle mean cost

Let $\mathcal G = (\mathcal V, \mathcal A)$ be directed graph with associated edge costs $c_{i,j}$ that has at least one directed cycle. Define the directed cycle mean cost to be $\frac {\{\text {sum ...
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1answer
68 views

Introduction to Linear Optimization: Driving the artificial variables out of the basis (case: no entries in the $j$-row are nonzero)

Reading the book Introduction to Linear Optimization by Bertsimas and Tsiklisis, I've come across the following subject: Driving the artificial variables out of the basis. The description is as ...
2
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1answer
74 views

How can the max-flow and min-cut problems, if dual to one another, both have unbounded optimal value?

The max-flow min-cut theorem states that the value of the maximum flow is equal to the minimum cut capacity. It is possible that the max-flow and min-cut is equal to $\infty$. However, reading ...
3
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1answer
64 views

Dual problem of a maximization primal problem $P$?

Suppose we have a primal problem $P$ which is stated as a maximization problem $\max c^{T} x$. The dual problem is defined (Introduction to Linear Optimization by Dimitris Bertsimas) only for a ...
2
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1answer
70 views

Assuming finite optimal cost of a specific LP, find an optimal solution directly

Minimize $\sum^n_{i=1} c_i x_i$ subject to $\sum^n_{i=1} a_i x_i = b$ (a single constraint), $x_i \ge 0$. Derive a simple test for feasibility of this problem Assuming the optimal cost is ...
5
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1answer
80 views

Finding a perfect matching using an LP

I have a basic question about the power of Linear Programming that has been bothering me for some time. I believe there is something simple I am missing. Linear Programming is $\mathsf{P}$-complete, ...
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1answer
96 views

Feasible solution existence

I wonder what is the fastest way to check whether the intersection of a set of half-spaces is empty. Right now I'm using a linear programming formulation (with Gurobi as solver) to check if there is ...
4
votes
2answers
444 views

Proof of Strong Duality Via Farkas Lemma

I am trying to prove what is often titled the strong duality theorem. There is a hint in the book that I'm following, and I want to follow the method they have outlined for me. I will outline the ...
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1answer
75 views

Check constraint under some condition in linear programming

I would like to minimize linear pseudo-boolean function $$\mathrm{obj} = \sum_i c_i \mathrm{sel}_i$$ subject to $$\sum_i c_i sel_i \geq \mathrm{Value} \qquad\qquad(1)$$ where $c_1,\dots c_5, ...
2
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1answer
465 views

Advantage of MTZ problem formulation of TSP

In class, we saw the Miller-Tucker-Zemlin formulation of the Travelling Salesmen Problem (TSP). MTZ is a way of formulating the TSP as an integer linear programming instance. I understand how MTZ ...
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0answers
53 views

What is linear relaxation in the context of bayesian networks? [closed]

To add onto the question, how are elliptical differential equations applicable in this context? I was listening to a talk about Bayesian networks and someone asked if they were using differential ...
3
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1answer
146 views

Boat riddle as a combinatorial optimization problem?

I remember a riddle about a bunch of people on a river bank, and a boat with limited capacity (lets say the boat can transport 2 people at a time). There are various relations between the people like: ...
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2answers
219 views

Applications for boolean logic operations in zero-one integer linear programming (ILP) [closed]

It is nice to know that every boolean formula can be expressed by zero-one integer programming by this answered question. But are there any applications? To be more precise: Are there papers which ...