All Questions
Tagged with linear-optimization or linear-programming
493 questions
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Family of functions which satisfies $f(\boldsymbol{x}) = 0$ if $\nabla f(\boldsymbol{x})=0$? [closed]
I have a Lagrangian of which I want to find the supremum in the primal variable $\boldsymbol{x}$:
$\mathscr{L}(\boldsymbol{x},\boldsymbol{\lambda})=f(\boldsymbol{x})^T\boldsymbol{a} + \boldsymbol{\...
1
vote
0
answers
47
views
Linear programs [closed]
Can the optimal value of the primal problem of a linear program ever be less then zero?
An example is: minimize $C=2x_1 +3x_2$ Subject to: $3x_1+4x_2 \leq 5$. Obviously, $x_1$ and $x_2$ are free ...
- 993
1
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0
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93
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quick hull algorithm detail
When using quick hull algorithm to find the polytope for half space intersection, we are required to provide an interior point to the solver qhalf.
In other words, providing
$$Ax \le b$$
is not ...
- 867
3
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2
answers
1k
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Equality constraints in mixed-integer optimization
Suppose I have a linear mixed-integer optimization problem of the form
$$MIP: min_{(x,y) \in \mathbb{R}^n \times \mathbb{Z}^m} c^\top x + d^\top y \hspace{0.2cm} \text{s.t.} \hspace{0.1cm} Ax+By \leq ...
- 342
1
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0
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43
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a question about probabilities on spaces of digraphs
Let $G$ be a directed graph with fixed nodes $s$ and $t$. Assume that each edge $e$ in the graph comes with a number $n(e)\in[0,1]$.
We consider probability spaces $S$ whose points are directed ...
- 61
4
votes
1
answer
966
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Solving for Hamiltonian path with constraints on allowable routes through vertices
Suppose you have a complete graph with N vertexes, with a distinguished vertex $n=1$ ("start"), and you wish to find a route traveling exactly once through each vertex so that the distance along the ...
- 151k
0
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1
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79
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algorithms and tools available for a particular polytope computation
Let me define each half space i as:
$${H_i}:{c_i}{\bf{x}} \le {b_i}$$
The intersection of all such ${H_i}$ gives a polyhedron (bounded or not). Suppose I am interested in if ${H_i}$ is active (...
- 13.6k
4
votes
1
answer
3k
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optimization of inverse matrix with constraint on matrix elements
everyone! I have this optimization problem with constraint.
$D$ and $T$ are symmetric matrices, where T is known and D is the unknown parameter.
$x$ and $v$ are two known p-dimensional vectors.
The ...
CommunityBot
- 1
3
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4
answers
4k
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Existence of nonnegative solutions to an underdetermined system of linear equations
Similar questions have been asked elsewhere, but I think this is sufficiently different to warrant a new post. I have a particular matrix $A$ and would like to know when the system $Ax = 0$ has at ...
1
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1
answer
73
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minimize number of unique elements in a vector
I was wondering if there is a simple or known way to minimize the number of unique elements in a decision variable (vector). Note that I'm not asking for minimization of nonzero elements (rank ...
- 14k
2
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2
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403
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Is this a linear optimization problem? $Ax=0$, $A$ has $m$ rows and $n$ columns, $m \le n$, all entries of $x$ are non-negative
$Ax=0$, $A$ has $m$ rows and $n$ columns, $m \le n$, all entries of $x$ are non-negative.
What should $A$ satisfy to guarantee the equation set have only zero solution?
27
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5
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2k
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Is the matrix $\left({2m\choose 2j-i}\right)_{i,j=1}^{2m-1}$ nonsingular?
Suppose we have a $(2m-1) \times (2m-1)$ matrix defined as follows:
$$\left({2m\choose 2j-i}\right)_{i,j=1}^{2m-1}.$$
For example, if $m=3$, the matrix is
$$\begin{pmatrix}6 & 20 & 6& 0 ...
- 43.2k
1
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0
answers
628
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Totally unimodular Matrices
A matrix is totally uni-modular if the determinant of any (square) sub-matrix is {+1, 0, -1}. My question is, "Is there a way to transform(linear or non) a general matrix into a totally uni-modular ...
9
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1
answer
3k
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Inverse of a totally unimodular matrix
A unimodular matrix $M$ is a square integer matrix having determinant $+1$ or $−1$.
A totally unimodular matrix (TU matrix) is a matrix for which every square non-singular submatrix is unimodular. A ...
9
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1
answer
6k
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Proving that a binary matrix is totally unimodular
I'm working on a set of problems for which I can formulate binary integer programs. When I solve the linear relaxations of these problems, I always get integer solutions. I would like to prove that ...
1
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1
answer
206
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Show $0-1$ Knapsack is polynomially reducible to this problem
I have already posted this question here but have not received an answer so I am cross-posting with hope to reach a larger amount of mathematicians:
Let $T=\{1,\cdots,n\}$ and consider the ...
CommunityBot
- 1
1
vote
1
answer
113
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Are convex combinations of 0-1 Pareto efficient vectors efficient?
Let $Y$ be any subset of $\{0,1\}^n$ for $n\geq3$. A vector $\alpha\in$ $Y$ is Pareto efficient if there is no $\beta\in$ $Y$ such that $\beta_i$ $\geq$ $\alpha_i$ for each $i\in\{1,...,n\}$ and $\...
- 1,502
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votes
2
answers
1k
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Linear programming is continuous
Consider an arbitrary linear program:
$$\max \vec c \cdot \vec x$$
subject to:
$$\textbf{A}\cdot \vec x = 0, \quad \vec a \le \vec x \le \vec b$$
Assume that this program is feasible and bounded. ...
- 884
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1
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81
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Can convex combinations of indicator functions for pairwise non-disjoint sets unordered by inclusion dominate one another?
Let $N$ be a finite subset of the naturals. Let $P$ be a set of subsets of $N$ such that:
1) $P\neq \varnothing$,
2) $\forall x\in P, |x| >1$,
3) $\forall x,y\in P,$ if $x\neq y$, then $x\not\...
- 13.4k
0
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0
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68
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A seemingly easy integer programming question
Let $k, m \in \mathbb{Z}_{ > 1}$. Let $a \in \mathbb{Z}_{> 0}^m$ and $t \in \mathbb{Z}^k$. Let $\varepsilon = (\varepsilon_{i,j})_{1 \leq i \leq m \\1 \leq j \leq k}$ be a matrix with entries in ...
- 501
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2
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246
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Are sums of 0-1 Pareto efficient vectors Pareto efficient?
Does there exist $m,n\ge1$, an $m \times n$ matrix $A$, and a vector $x \in \mathbb{R}^n$ such that:
The entries of $A$ are $\in \{0, 1\}$.
For all pairs of columns $u, v$ of $A$ the entries of $u - ...
- 11
1
vote
1
answer
184
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Do doubly infeasible Linear Programming problems always have doubly infeasible bases?
Consider a Linear Programming problem in dictionary form,
$$\max\Big\{f^\pi+\!\!\!\sum_{j\in D(\pi)}\!\! d^\pi_jx_j~\Big|~\forall~i\!\in\!B(\pi)~~~ b^\pi_i+\!\!\!\sum_{j\in D(\pi)}\!\! G^\pi_{ij}x_j\...
CommunityBot
- 1
2
votes
1
answer
3k
views
max-flow at max-cost
I have a flow network with gains. In practical terms, a gain is the opposite of a cost. So, I interested in finding the maximal gain of a network flow, what could be interpreted as finding a maximum ...
CommunityBot
- 1
5
votes
3
answers
8k
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Linear programming - uniqueness of optimal solution
Is it possible to build such an objective function for a given set of constraints, so that there will be only one optimal solution?
My general problem is to get any vertex of a polytope formed by a ...
- 367
2
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2
answers
3k
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Linear programming with infinitely many constraints
I wish to study the following linear program
$$\begin{array}{ll} \text{minimize} & \mathrm c^{\top} \mathrm x\\ \text{subject to} & \mathrm A \mathrm x = \mathrm b\\ & \mathrm x \geq 0\...
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2
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120
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Reference request: dependence on linear constraints
Excuse me if my question is stupid. I'm seeking the references on the dependence of the (linear) optimization problem on (linear) constraints. Namely, consdier the following optimization problem:
$$P(...
- 1,835
3
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0
answers
71
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Dependence of optimization problem on the linear constraints
Let $I=\{x_1,\cdots, x_n\}\subset \mathbb R$ be fixed. Given two probability distributions $\alpha=(\alpha_i)_{1\le i\le n}$ and $\beta=(\beta_i)_{1\le i\le n}$ on $I$, and a matrix $c=(c_{i,j})_{1\le ...
- 1,835
2
votes
1
answer
1k
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Complementary slackness for approximately optimal Dual solution
Given a Primal LP (P) and it Dual LP (D) we know that the optimal solutions to P ($x_{opt}$) and D $(y_{opt})$ satisfy complementary slackness condition, i.e. under optimal solutions either a ...
- 54.2k
0
votes
0
answers
890
views
Maximum shortest path problem
I have the following problem. You have a graph and every edge has a certain set of possible weights. The question is to find the assignment of those weight which will maximize the shortest path.
In ...
- 342
2
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0
answers
126
views
Unveiling hidden structures
One way to unveil a hidden structure of a undirected graph - given as an adjacency matrix - is to permute the rows and columns until a pattern with a maximal geometrical symmetry is found. (The ...
CommunityBot
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1
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0
answers
1k
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Number of different combinations in a 0-1 knapsack problem with integer weights [closed]
My question is actually very similar to this other one: Given a vector of positive integers, count the number of combinations which have a sum that produces a different value. But, since this previous ...
CommunityBot
- 1
3
votes
1
answer
634
views
Properties of one dimensional null space
Let $\mathcal{G}$ be denote the set of all $3 \times 3$ real symmetric matrices and let $\mathcal{G}^+$ denote the set of all $3 \times 3$ positive semidefinite matrices (see definition).
Let $S: \...
- 1
4
votes
2
answers
722
views
Minimum number of rectangles in a polygon
Given a polygon and dimension $d$, find a minimum partition of rectangles that has either of its dimensions equal to $d$.
Example:
Consider the following diagram:
I want to cover maximum shaded ...
- 13k
3
votes
2
answers
1k
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SDP relaxation vs LP relaxation
I have a question I hope you might be able to answer.
Let's say we have an integer program for the stable set problem (or clique, not principal).
\begin{equation}
\begin{aligned}
& \text{...
CommunityBot
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1
vote
0
answers
1k
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Analytic formula for minimizing the maximum inner product of a set of vectors
Given $x_j\in\mathbb{R}^n$, $j=1,\ldots,p$, find
$$
\widehat{w} \in \arg\min_{\Vert w\Vert=1}\max_{1\le j\le p} |\langle w,x_j\rangle|.
$$
I am also interested in the special case where we further ...
- 710
0
votes
1
answer
201
views
Recursive linear programming on a linear subset of a simplex
The problem I am working on is:
Given an $n$ dimensional vector $r \in \mathcal{R}^n$, and a convex set $G=\{\mu \in \mathcal{R}^n | \mu_i \ge 0, ~ \mu^T \mathbf{1}=1, ~ A\mu =0 \}$ where $\mathbf{1}...
- 61.9k
2
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3
answers
1k
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Quadratic Programming With Piecewise Linear Term
The problem I have can be defined as:
$$
\min \frac{1}{2}\mathbf{x}^T\mathbf{Q}\mathbf{x} + \mathbf{c}^T\mathbf{x}
$$
s.t. linear equality constraints:
$$
\mathbf{Ax=b}
$$
and linear inequality ...
- 101
0
votes
1
answer
212
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How to find out if a polytope contains a sphere?
Given a polytope described by linear inequalities $Ax \le b, x \in \mathbb R^n$, how do you find out if there exist a (non degenerate) sphere of dimension $n-1$ contained in the polytope?
Thanks!
- 54.2k
5
votes
1
answer
146
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How does one go from convexity to submodularity?
If I have a function which is convex in the hypercube, $[-1,1]^n$ then when would it imply that its restriction to $\{-1,1\}^n$ is submodular?
It would be helpful is someone can share some specific ...
- 111
3
votes
2
answers
2k
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Sherali-Adams relaxation
I am trying to find a book or a paper, which explains, how and why the Sherali-Adams relaxation method works. The original paper (1990) is difficult for me to understand. I need a more basic ...
2
votes
1
answer
529
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Integer programming and Groebner basis
I enjoyed reading different papers about using Groebner basis to solve integer programming.
Is there any literature about the complexity and/or comparison with other (more classical) methods like ...
- 10.9k
1
vote
0
answers
187
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Strong Duality of Mixed Integer Linear Program
The problem at hand is to optimize a mixed-integer linear program closely related to the maximum flow problem. I would like to reformulate the problem with its dual and I'm concerned with the ...
- 11
2
votes
1
answer
171
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Maximization of Binary Multilinear Fractional Function
Problem: Let $a_{i,j}$, $b_{i,j}\in\mathbb{R}$ for all $(i,j)\in\left[m\right]^2$ such that $a_{i,j}=a_{j,i}$ and $b_{i,j}=b_{j,i}$. Let $z_k\in\{0,1\}$ for $k\in\left[m\right]$. We wish to maximize,
...
- 682
1
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1
answer
155
views
Derive a vertex representation of a permutohedron from its linear-inequalities form
Let us define the $n$-permutohedron $P_n$ as the set of all $x\in\mathbb{Q}^n$ such that
$$\sum_{i=1}^n x_i = \binom{n{+}1}{2}\ \ \ \land\ \ \ \forall\,\text{nonempty}\ S\subsetneq\mathbb{N}_n\colon\ ...
- 27.8k
1
vote
1
answer
1k
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convert absolute form into linear programming problem [closed]
I would like to convert this problem into a Linear Programming Problem :
$\min |x|+|y|+|z|$
subject to $x+y \leq 1$
$2x+z=3$.
The solution to this problem is given chapter and here. But I still ...
- 146
4
votes
1
answer
345
views
Existence of Nonnegative Solutions of Linear Systems of Equations and Inequalities with particular constraints
Suppose we have an $n \times m$ nonnegative matrix $A$, where each row sums to $1$. I wonder whether there exists an $m \times n$ nonnegative matrix $X$ that satisfies the following constraints:
...
CommunityBot
- 1
1
vote
0
answers
64
views
Maximize discrete harmonic function at given point
Let $n>0$, and let $S_n$ denote the discrete square
$S_n=[|-n,n|]^2$ (so $S_n$ has $(2n+1)^2$ elements). Let $K_n$ denote the set of four corner points $\lbrace (\pm n,\pm n)\rbrace$, and $C_n=S_n\...
- 621
2
votes
2
answers
438
views
Perturbation of Linear Programs
Consider the linear program,
$$\begin{array}{ll} \text{maximize} & c^T x\\ \text{subject to} & Ax \leq b\\
& x \geq 0\end{array}$$
I want to study the sensitivity of the optimal $x^*$ ...
- 313
5
votes
2
answers
2k
views
Bounding the minimal maximum norm of a solution of a linear system.
I would be grateful for pointing me out a reference to some general bound on the $\ell_{\infty}$ norm of a solution of a linear system. To be specific, suppose that we have an underdetermined linear ...
- 791
1
vote
0
answers
55
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Separation on discrete set
Consider the set $L = \prod_{i=1}^n\{1,0\}$, i.e. L consists of the element of n-tuples whose entries are 0 or 1. Also we can regard $L$ as a subset of $R^n$.
Define linear functions $f(x)= a_1x_1+ \...
- 61