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

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Project to nearest point in convex polytope

Is there a reasonably efficient algorithm for the following task? Input: a point $x \in \mathbb{R}^d$; a convex polytope $\mathcal{C} \subseteq \mathbb{R}^d$ Find: a point $y \in \mathcal{C}$ that is ...
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1answer
17 views

Does there always exist equivalent (M)(I)LPs with and without objective functions?

For computing pure Nash equilibria (game theory), there exists a MILP method in literature (clicky). In the proposed MILP, there is no objective function. A solution is a pure Nash equilibrium if it ...
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1answer
31 views

Flaw in linear programming solution for multi-commodity flow problem?

The multi-commodity flow problem problem statement - wiki According to constraints of multi-commodity flow problem a given material must start at source s with demand d and end up at its target t. ...
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44 views

Reducibility of finding Eulerian Path to Linear Programming

Consider any arbitrary directed, acyclic graph; how can we formulate the problem of finding a particular Eulerian path as a linear programming problem? It seems like there should be a relatively ...
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31 views

How to construct a network flow problem?

I have the optimization problem given below max $\sum_{i=1}^{N}\sum_{j=1}^{M} x_{ij}R_{ij}$ s.t $\quad 1)\quad \sum_{j=1}^{M} x_{ij}=1 \quad \forall i$ $\quad 2)\quad x_{ij} \in {0,1}$ $\quad ...
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31 views

Exponential example for simplex used in SMT solvers

The original simplex requires an exponential number of pivot operations -- e.g., if run on the Klee-Minty example [3,4]. What about the simplex algorithm used in SMT solvers [1,2]: Could you provide ...
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35 views

Maximum set packing and minimum set cover duality

I read that the maximum set packing and the minimum set cover problems are dual of each other when formulated as linear programming problems. By the strong duality theorem, the optimal solution to the ...
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12 views

Writing linear programming constraint in a canonical form

I have a particular research problem that I'm formulating as a linear program. It's more or less an instance of the transportation problem, except there is one additional constraint that is proving ...
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71 views

Dynamic Shortest Path with Linear Programming

Consider a grid with $x=5$ columns, $y=5$ rows, and $T$ timesteps. There are $N=2$ agents in this grid, which can move vertically or horizontally. The positions of each agent $x$ at timestep $t$ ...
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44 views

Intuitive self-contained proof of Farkas' Lemma

I've been studying the proof of Farkas' Lemma, and given my rather fuzzy memory of Linear Algebra, am having some trouble with it. One version of Farkas' lemma states: For any convex cone ...
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29 views

Showing a linear program is infeasible or finding a feasible solution

I'm aware that for any given maximize/minimize LP problem, if its dual is unbounded then the primary is infeasible and vice versa. But what if there is no maximize/minimize objective function? For ...
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35 views

How to prove a min-max and max-min strong duality for Linear Programming example?

I was practicing Linear Programming, and came across the following question: Let $G = (V, E)$ be an undirected graph with capacities $c_e$ on each edge $e \in E$. let $s$, $t$ be two of its vertices. ...
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51 views

Max Flow / Linear Programming Reduction Variant

While studying max flow / LP, I came across a couple of reduction problems that gave me a bit of pause: Here are two variants of the standard Maximum Flow problem. Show that both of them can be ...
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70 views

General Steiner Tree Variants

In the general Steiner tree problem (Steiner tree in graphs), we are given an edge-weighted graph G = (V, E, w) and a subset S ⊆ V of required vertices. A Steiner tree is a tree in G that spans all ...
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1answer
37 views

Express product in ILP

Suppose I have a mixed integer-linear program (MILP) with variables $x,y,z$, where $y$ is a 0-or-1 variable, and I want to impose the constraint $z=xy$. This is not expressible in a MILP directly. ...
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37 views

Solution to a Np-hard problem and its relevance to a dual LP

From The design of APX algorithms book by David P. Williamson and David B. Shmoys, at the bottom of page 21 I saw the following statement (it is about the set cover LP and its dual): Let $y^*$ be ...
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1answer
55 views

Use max operation in a constraint in Linear Programming

I have liner programme with set of $x_{3n}$ variables where $x_{ij}$ are {0,1}. I am solving this linear programme using LP-Solve. Using these variables, I want to form following constraint : $max(...
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1answer
87 views

Single-source shortest paths as a linear program

I saw that I can formulate single-source shortest path as the following linear program: Given $G=(V,E)$ and $w\colon E\to R$ and with negative cycles, find $\max\,d(s,t)$ such that \begin{align*} ...
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75 views

Algorithm to optimize polling frequency between producer and consumer

I am trying to optimize what we call AJAX request polling frequency in the domain of web design. Here's a general version of the problem in simple lingo: Problem Statement: Suppose there are 3 ...
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86 views

What's the complexity of solving a packing LP?

Linear Programming is in polynomial time weakly (when numbers are encoded in unary). AFAIK it remains open if it is possible to solve LP in polynomial time strongly (when numbers are encoded in ...
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1answer
54 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
49 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 ...
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76 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 ...
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65 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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76 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 ...
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78 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 ...
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39 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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41 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 ...
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110 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 \alpha(x)V(x)\\...
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3answers
236 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 $x,...
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1answer
111 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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47 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 ...
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147 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 ...
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24 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 ...
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1answer
52 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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34 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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50 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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38 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 ...
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1answer
100 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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49 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 ...
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395 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: ...
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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 \vec{...
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129 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 ...
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1answer
79 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 ...
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1answer
53 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 ...
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39 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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179 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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102 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 ...
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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 ...
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21 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 ...