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Questions tagged [linear-programming]

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

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Reducing linear programming to positive linear programming

Suppose we have an oracle that solves problems of the form \begin{align*} \text{maximize} ~~ & c^T x \\ \text{subject to} ~~ & A x = b, x\geq 0 \end{align*} when $c\geq 0$ (all coefficients in the maximization target are non-negative). Can it be…
Erel Segal-Halevi
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How does cycling happen in the simplex method?

I'm reading Schrijver's Theory of Linear and Integer Programming, and I have a problem understanding cycling happens in the simplex method. The simplex is described as below: Solving $\max\{cx\mid x \geq 0; Ax \leq b\}$ At each stage a set $C_k$ is…
J. Doe
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Use complementary slackness to prove the LP formulation of max-flow only need polynomial number of path constraints

This is a homework problem for a class that ended 2 years ago, I'm learning it by myself. Consider a directed graph $D=(V,A)$, $s,t\in V$. $A=\{a_1,\ldots,a_n\}$. Let $P=\{p_1,\ldots,p_m\}$ be the set of all simple paths from $s$ to $t$, There is a…
Chao Xu
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Need help optimizing the loading of passengers on small airplanes

I work for a small non-profit that provides transportation for people who need medical treatment. We connect volunteer private pilots who fly people in their own (small) aircraft, typically 3-5 seats. Several times each year, we provide flights for…
Stephan
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Linear programs with strict inequalities and supremum objectives

Linear programming can solve only problems with weak inequalities, such as "maximize $c x$ such that $A x \leq b$". This makes sense, since problems with strict inequality often do not have a solution. For example "maximize $x$ such that $x<5$" does…
Erel Segal-Halevi
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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 primal minimization problem. Then what is the dual…
Shuzheng
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Advantages of Integral over Non-integral Linear Program?

I have a linear program over real variables for which it can be shown that all the vertices of the polytope describing its feasible region are integral. Obviously I can just solve this using a standard linear programming solver. However I would like…
Richard Forrest
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Start simplex method from feasible internal point

I have one algorithm that generates a feasible solution to a linear programming problem. However, it is very likely that this is not a corner point. This makes it not suitable for direct use as an initial feasible solution for a bounded Simplex…
Dylan
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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? Can we make a drawing to see when we will have a…
Evinda
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The set of possible values of linear programs

Consider the set of all linear programs of the form: maximize $c x$ subject to $A x \leq b$ $x \geq 0$ where there are $m$ variables, $n$ constraints, and all coefficients in $A, b, c$ are integers from a given finite set $K$. Since the number of…
Erel Segal-Halevi
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Linear Program, minimize maximum distance

Given: Set $N = \{0, \ldots, n-1\}, k \in \mathbb{N}, d_{ij} \geq 0$ with $d_{ii} = 0$. Task: Find subset $C \subseteq N, |C| \leq k$ that minimizes $\max_{i \in N} \min_{j \in C} d_{ij}$. Idea: I define $x_i$ with $x_i = 1$ if $i \in C$ and zero…
Niah
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Lower bound on positive coefficients of the optimum of 0,1-linear programming problem

I have an instance linear programming such that the coefficients and the constant terms are 0 or 1. Formally, the set of variables is denoted as $V$ and $|V| = n$. There are $m$ constraints, formed as $$ \mathbf{a}_{1}^{\top} \mathbf{v} \leq 1…
Blanco
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Ensuring integral result from integral linear program

An integral linear program is one that has a maximizer that is integral. Sometimes it's possible to prove that a particular LP has this property, for example by proving that it's constraint matrix is totally unimodular. Suppose that we have an…
orlp
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Some Questions related to Linear programming

I have some question related to the Linear Programming problem: If we have an objective function that needs to be maximized and let the feasible region be unbounded such that there is no finite optimum solution. My question is what is the meaning…
John adams
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LP - Dual variable is zero implies primal constraint unnecessary?

Say I have a primal program P with n variables and c constraints. Let's say that I have an optimal solution for the dual program D, in which the y1, the variable related to the first constraint in P, is zero. Does that imply that I could run…
tkalir
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