All Questions
Tagged with linear-programming linear-programming or
492 questions
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96
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On optimizing a multivariate quadratic function subject to certain conditions
The problem is to maximize $f(x_1,x_2,\cdots,x_n)=\sum\limits_{i=1}^{n}\Big(x_i-k_i\Big)^2$ for $n\ge 3$ subject to the conditions (1) $\sum\limits_{i=1}^{n}x_i=\sum\limits_{i=1}^{n}k_i\le n(n-1)$ ...
- 61
1
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0
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335
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Closed-form solution of a particular linear program
(Note: I asked a similar question at math.stackexchange but the present one is more precise.)
I have a linear program of the form:
$$\text{minimize} \space\space x_1 \space\space \text{subject to:}$$
$...
- 4,219
1
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1
answer
169
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Best projection on non-convex discrete set with two constraints
I want to compute the projection of a vector $\left( x\right) _{1\leq
i,j\leq n}\in \lbrack 0,1]^{n\times n}$ on the following discrete set
$$
S=\left\{ x\in \{0,1\}^{n\times n}:x_{i,j}+x_{j,i}\leq 1;\...
- 128k
2
votes
1
answer
61
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Counting the number of pair of d-uplets with upper bounded distance
Consider two d-uplets $u = (u_1,...,u_d)$ and $v = (v_1, ..., v_d)$ both living in $\mathbb{N}^d$ with $d$ a positive integer. They both verify $$(*) \sum_{i=1}^d u_i = \sum_{i=1}^d v_i = k$$ with $k$ ...
0
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1
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147
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Is there a redundant constraint in linear programming? [closed]
From wikipedia:
But... Why do we need the $x\ge 0$ part? We can instead do $-x\le 0$, and thus saving a line in the definition (which is not a big deal but nevertheless nice).
(In order to do that, ...
- 63.9k
0
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0
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272
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Finding the eigenvectors of a submatrix
Let $A=(a_{kl})$ be a matrix in $M_n(\mathbb{R})$ when $n$ is even. Let $B=(b_{kl})$ be the symmetric $2n$ by $2n$ matrix whose entries are given by,
$b_{k,l}=a_{kl}$ if $1\leq k,l\leq n$.
$b_{n+k,l}=...
- 4,058
1
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0
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47
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Nash Equilibria change linearly in (some) game parameters. Already known / follows from a more general result?
EDIT: The key thing that I am wondering about is the linearity of the P2 strategy, not the constancy of P1. (The latter is straightforward.)
Question: Is the following result already known? Or is it a ...
0
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1
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82
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Combining Dantzig-Wolfe and Benders decomposition
I'm now solving an LP that has a few coupling rows (as in Dantzig-Wolfe decomposition) and a few coupling columns (as in Benders decomposition) simultaneously; other rows and columns are block-angular....
- 5,429
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1
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405
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Computing discrete optimal transport
I am trying to find a combinatorial approach to solve the following optimization problem.
\begin{align}
&\max_{x_{ij}} C_{ij} x_{ij}, \\
&\text{such that},\\
&\sum_{j} x_{ij} \leq r_i~\...
CommunityBot
- 1
0
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1
answer
36
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Benefit of adding a trivial constraint to ILPs
let ILP be an integer linear program with constraints-matrix $\boldsymbol{\mathrm{M}}\in\mathbb{Z}^{m\times n}$ and cost vector $\boldsymbol{\mathrm{c}}\in\mathbb{Z}^n$,
${\boldsymbol{\mathrm{x}}^*}\...
- 5,429
0
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1
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93
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How quickly can this IQP or its MILP relaxation be solved
Let $A\in\{0,1\}^{(n,n)}$ be a $n$ by $n$ boolean matrix (in particular think of an adjacency matrix of a graph), and consider the following optimization problem:
$$\begin{align*}&&\max_{P\in\{...
- 71
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94
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Boolean operation on n dimensional polyhedron
A polyhedron in $R^n$ is defined by a set of half-planes: $P = \{x \in R^n \mid Ax - b \le 0\}$.
Given a set of polyhedra in $R^n$, $ P_1, P_2, \dotsc, P_k$, is there an algorithm/implementation that ...
- 5,429
0
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1
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538
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Method for (binary) optimization under constraints
I would like to know if there is a method to solve the Problem.
Problem:
Maximize the following function: $$f(p_{1,i},p_{2,i},\dotsc,p_{m,i})=\sum_{i=1}^{n}\begin{bmatrix}p_{1,i} & p_{2,i} & \...
- 12.9k
1
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0
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59
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How do I incorporate Ito's lemma into the solution for a finite-horizon stochastic cake-eating problem?
I'm interested in finite-horizon, continuous-time cake-eating problems in which the agent has a time-horizon $W$ over which to eat the cake, and then chooses an optimal consumption path $\{h_t\}_0^W$, ...
- 11
2
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0
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299
views
Practical application of envelope theorem for linear programs
Assume that we have solved a (standard) linear program
$$
\text{minimize}_{x\in {\mathbb R^n}}\,\, c_0^Tx, \,\,\,\,\, \text{s.t. } A_0x \leq b_0,
$$
and would like to know how sensitive is the optimal ...
- 1
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1
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143
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$\mathrm{ILP}$-formulation for Minimum Maximal Matching (MMM) Problem
Despite some online searching I couldn't find examples of dedicated Integer Linear Programs ($\mathrm{ILP}$s) for determining smallest matchings, that are not contained in a larger one.
It seems that ...
- 5,429
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1
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64
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Round Robin volleyball Tournament [closed]
Consider a set of N teams (N even number) that must make a
Round Robin Tournament. To each pair i; j, i ≠ j, of teams there is associated level
of interest si,j ∈ {1;2;3} of the match between them (1 =...
- 5,429
2
votes
1
answer
227
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Solving linear programming without solving linear programming
Let $v_1, \cdots, v_n$ be vectors in $\mathbb R^k$, and let $M$ be the Gram matrix of them.
It's possible to determine from $M$ and $k$ whether the only vector that has nonnegative inner product with ...
- 9,501
2
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1
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372
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Who called Farkas' fundamental theorem a lemma?
Farkas proved his famous result (which, nowadays, is fundamental in optimization theory) in 1902 and called it Grundsatz der einfachen Ungleichung which may be translated as fundamental theorem of ...
- 188k
2
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1
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508
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Under what condition does Courant–Fischer–Weyl min-max principle hold in general?
From Wikipedia:
Let $A$ be an $n \times n$ Hermitian matrix. As with many other variational results on eigenvalues, one considers the Rayleigh–Ritz quotient $R_A :
\mathbf C^n \setminus \{0\} \to \...
- 12.9k
6
votes
2
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8k
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Existence/Uniqueness of Nonnegative Solutions of Linear Systems of Equations
Suppose we have an $m$x$n$ matrix $A$, with $m\lt n$, and an $m$x$1$ vector $b$. Are there existence and uniqueness conditions characterizing nonnegative solutions of the system of linear equations $...
7
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2
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1k
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Is a given point in the interior of the convex hull of a given finite collection of points?
Suppose I have the convex hull $P$ of a finite collection of points in $\mathbb{R}^d,$ and I want to see whether a point $p$ is contained in $P.$ This is a standard (some would say the standard linear ...
0
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1
answer
319
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Correct way to conduct equilibrium scaling of linear/integer/MIP program
I would like to scale my linear/integer program and also mixed-integer program using the equilibrium scaling method. I have worked on two research papers and one research book. However, they did the ...
- 21
4
votes
2
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314
views
Connecting $2n$ points in $\mathbb R^2$ with line segments s.t. each point belongs to exactly one line segment
I'm trying to do a certain simulation related to the toric code and I'm looking for an algorithm that connects $2n$ points ($n \in \mathbb Z_+$) in $\mathbb R^2$ with line segments with the following ...
- 5,429
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61
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Linear programming robustness to input perturbations
I'm running a linear program whose parametrization depends on the output of a neural network. I was wondering if there exist results on how robust linear programs are towards perturbations in their ...
- 11
2
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1
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644
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How to maximise infinity norm of $x$ with constraint $Ax \le b$ using linear program? [closed]
I want to maximise the infinity norm of $x$, subject to constraint: $Ax \le b$. I think you can use a linear program to solve this, but how do you go about formulating it?
- 162
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0
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115
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Explicit equation for border of the Minkowski sum of sets
Assume we have sets of the form
$$
M_j = \{x\in\mathbb{R}^d : f_j(x) \le 0,x \ge 0\}
$$
where $x\ge 0$ means $x_i \ge 0 \quad \forall i=1,\dots, d$.
Goal
I am looking for an (explicit) representation ...
- 63.9k
1
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0
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98
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Solution of a simple optimization problem
Let $\mathbf{U}_1$ and $\mathbf{U}_2$ be two arbitrary unitary matrices and $\mathbf{D}$ be a diagonal matrix. What is the solution of the following optimization problem?
\begin{align}
\min_{\mathbf{...
- 63.9k
0
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0
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124
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The best unitary matrices that approximate a matrix product
Let $\mathbf{A}$ be an arbitrary $N\times N$ complex matrix. Moreover, $\mathcal{U}_1$ and $\mathcal{U}_2$ are distinct subsets of all unitary matrices. Suppose the matrices $\mathbf{U}_1$ and $\...
- 287
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35
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How to chose the start vector for the MTZ variables
In the context of LP-formulations for the Traveling Salesman Problem the MTZ constraints prevent subtours via $n$ (i.e. effectively $n-1$) additional variables $$u_1=1\\2\le u_2,\,\dots ,\,u_n\le n\\ ...
- 4,713
1
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0
answers
200
views
Drawing a 3D object in a 3D environment, and converting to math [closed]
So I have been granted a free time and I want to work on a project but first I had to research.
As we know, lines have infinite points, and with lines, we can create infinite shapes. I want to let ...
- 11
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0
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108
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How to find a set given its support function
Let $\mathcal{U}$ be a convex and compact set. Its support function is defined as $\delta^*(v|\mathcal{U})=\sup_{u\in \mathcal{U}} v^T u$. Assume that we are given the support function $\delta^*(v|\...
- 53
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0
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40
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Subtour-gluing constraints for ILP formulation of TSPs
If one doesn't want to introduce additional variables to the ILP of a TSP instance, one has to add exponentially many so-called subtour-elimination constraints; in practical calculations subtour-...
- 13.2k
0
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0
answers
96
views
Why is Gaussian distribution always chosen for smoothed analysis?
I came across the algorithmic perfomance analysis model of smoothed analysis. In all references that I read a Gaussian distribution was used for perturbation (e.g. Spielman and Teng 2004 for the ...
- 29
1
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1
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82
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Do we really need degree constraints for ILP formulations of TSP problems
The Dantzig-Fulkerson ILP-formulation of the symmetric TSP is
$$\min\sum\limits_{i=1}^{n-1}\sum\limits_{j=i+1}^n c_{ij}x_{\lbrace i,j\rbrace}\quad\text{s.t.}\\ \sum\limits_{j\ne i,\,j=1}^n x_{\lbrace ...
- 5,429
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58
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Second-order envelope theorem for linear programming
Consider parameterized linear programming $V(\theta) = \max_x \langle c(\theta),x\rangle$ s.t. $A(\theta)x\leq b(\theta)$, $x\geq 0$. Let's also assume $c,A,b$ are infinitely differentiable with ...
- 373
1
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1
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475
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Sufficient conditions for a system of linear inequalities to admit a solution
I am looking for sufficient conditions such that a system of linear inequalities of the type $A x >0$ admits a non-negative solution $x \in \mathbb{R}^n_+$. I know a few properties of the $m \times ...
1
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0
answers
37
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Sum of all integer binary solutions of a TUM linear system
I have the following problem: $A x = b$ where $A$ is a $m \times n$ total unimodular matrix (TUM) with entries in $\{0,1\}$ and $b$ is a $m$-vector of strictly positive integers. Let $\mathcal X$ be ...
- 11
4
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1
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866
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When is a triangular matrix totally unimodular?
I have a {0,1}, invertible, triangular matrix, that I would like to show is totally unimodular. Are there any known results on the total unimodularity of classes of triangular matrices?
- 32.1k
2
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2
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418
views
Lovasz theta function - uses
Lovasz theta function bounds the Shannon capacity of graphs. What are some other uses of the function - especially in asymptotic coding theory and optimization problems?
- 32.1k
16
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3
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1k
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Can a convex polytope with $f$ facets have more than $f$ facets when projected into $\mathbb{R}^2$?
Let $P$ be a convex polytope in $\mathbb{R}^d$ with $n$ vertices and $f$ facets.
Let $\text{Proj}(P)$ denote the projection of $P$ into $\mathbb{R}^2$.
Can $\text{Proj}(P)$ have more than $f$ facets?
...
- 32.1k
2
votes
0
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76
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Polyhedron coordinate bound
Given a polyhedron
$$Ax\leq b$$
where we assume $A\in\mathbb Q^{m\times n}$ and $b\in\mathbb Q^{m}$ and it takes $L$ bits to represent the inequalities what is a good bound on the quantity $\|y\|_\...
- 13.9k
0
votes
0
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165
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Minimum circumscribed ellipsoid of $\mathcal H$-polytope
Given matrix $A \in \mathbb{R}^{m \times n}$ and vector $b \in \mathbb{R}^n$, consider the $\mathcal H$-polytope $P$ defined as follows
$$ P := \left\{ x \in \mathbb{R}^n : Ax \leq b \right\} $$
I ...
0
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0
answers
137
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Any technique for linearization, or linear approximation?
Consider the following Matrix constraint:
$$
\begin{bmatrix} -U+\psi\Sigma_b^{-1} & V \\ V^T & -V^TU^{-1}V+\tau_2 -\psi \end{bmatrix} \leq 0
$$
where $\Sigma_b$ is a known positive definite ...
0
votes
1
answer
131
views
How hard is a linear programming with a bounded constraint?
Background: I am reading Greg Kuperberg's answer to the question Deciding membership in a convex hull. I am thinking about the complexity of ''Deciding membership in a convex hull''.
Restate the ...
- 63.9k
3
votes
0
answers
282
views
Continuum of Lagrange multipliers, duality gap, and minimax theorem
Suppose I have a linear optimization problem involving random variables on some (infinite) probability space $\Omega$. For example, need to maximize expectation $E[Q]$ of random variable $Q$ subject ...
- 781
0
votes
1
answer
213
views
How do you call a linear programming problem when the solution should be "constrained" to a norm?
(apologies for the n00b question)
Let's say we have a vector of length $n$, with to-be-determined values: $a_1, a_2, ...,a_n$.
And we have information that partial sums of these elements are equal to ...
- 1,714
0
votes
1
answer
110
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Detecting non-negativity of a single constraint by polyhedral constraints - $I$
We consider $$\langle a,x\rangle=b$$ (linear constraints) where $x\in\mathbb R_{\geq0}^n$ and every entry in $a=(a_1,\dots,a_n)$ is in $\mathbb Z_{\geq0}^{n}$ (non-negative) and the entry $b$ is in $\...
- 14k
1
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0
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162
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Optimization problem on trace of complex matrix product
Given a complex rectangular matrix $A$ $(k \times n)$, I am interested in solving the following optimization problem over $(k\times n)$ complex matrices $x$:
$$
\mathrm{arg}\max_X \,\mathrm{trace}(X^...
- 377
1
vote
0
answers
43
views
Detecting non-negativity of a single constraint by polyhedral constraints - $II$
Let $$\langle a,x\rangle=b$$ be a linear constraint where $x\in\mathbb R^n$ and every entry in $a=(a_1,\dots,a_n)$ is in $\mathbb Z_{\geq0}^{n}$ (non-negative) and the entry $b$ is in $\mathbb Z_{\...
- 13.9k