Linear programming is the study of optimizing a linear function over a set of linear inequalities. The Simplex Method, Ellipsoid Method and Interior Point Method are popular algorithms to solve linear programs.

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Does it make sense to compare sets (polytopes) with different dimensions? [on hold]

In the context of integer programming, I am considering 3 different linear models for a given problem. The goal is to determine which formulation is the tightest, that is, the one whose polytope is ...
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51 views

Existence of Nonnegative Solutions of Linear Systems of Equations and Inequalities with particular constraints

Suppose we have an n by m nonnegative matrix A, where each row sums to 1. I wonder whether there exists an m by n nonnegative matrix X that satisfies the following constraints: each row of X sums to ...
2
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1answer
56 views

Minimum cover for sets in which each element appears in exactly 2 sets?

Is there an algorithm for finding minimal covers of a set of sets in which each element of the universe appears in exactly 2 sets? I realize that LP relaxation approximates this to within a factor of ...
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57 views

Optimization with vectors

I am trying to solve the following optimization problem as a small part of a research project, and I do not know if there exists closed form solutions. My linear algebra is very rusty and I am looking ...
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47 views

About identifying a few diagrams

Please have a look at these beautiful seminar slides, https://math.berkeley.edu/~bernd/coimbra1.pdf Can someone kindly identify the algebraic description of the spectrahedron that is drawn on slide ...
4
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128 views

Closed-form solution of a linear programming question

Among all the probability matrices \begin{equation*} P = \left(\begin{array}{cccc} p_{00} & p_{01} & \ldots & p_{0,J-1} \\ p_{10} & p_{11} & \ldots & p_{1,J-1} \\ \vdots & ...
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1answer
42 views

Perturbation of Linear Programs

Consider the linear program, \begin{align} & \max c^Tx \\ & \mbox{such that }Ax \leq b \\ & \mbox{ and } x \geq 0 \end{align} I want to study the sensitivity of the optimal $x^*$ ...
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27 views

Maximisation of a discrete linear function

I am trying to maximise the function $$ Q=\sum_i{\alpha _{i}x_{i}} $$ subject to the constraint $$ W<=\sum_i{\alpha _{i}w_{i}x_{i}} $$ By changing $\alpha _{i}$ subject to $ 0<=\alpha ...
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26 views

Equivalence between multiclass SVMs, power diagrams, and constrained $k$-means

Apologies in advance for the long post: Suppose we have a collection of points $\mathbf{p}_1,\dots,\mathbf{p}_n$ in $\mathbb{R}^d$, and we consider the following three ways of partitioning these ...
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1answer
99 views

Is the linear production game a convex game?

In cooperative game theory, the linear production game (LPG) is defined by letting the characteristic function have the form of a linear programming problem. Does anyone know if the LPG is a convex ...
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1answer
60 views

Efficiently Generating the Convex Hulls of Two Polytopes and Counting Faces

Suppose you have two polytopes $P_1, P_2 \in \Bbb{R}^n$ given by $$ P_1 = \lbrace x: A_1 x \le b_1\rbrace$$ $$ P_2 = \lbrace x: A_2 x \le b_2\rbrace $$ I wish to find their convex hull, that is a ...
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30 views

Linear Program for Single Source Shortest Paths Tree

This question originates in quick, however wrong, idea to calculate a shortest paths tree in the presence of negative cycles. The essential motivation was that a linear program would determine binary ...
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110 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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19 views

Is there a standard technique for overdetermined homogeneous integer programming?

Given $M\in\Bbb Z^{n\times n}$ of $\mathsf{rank}(M)=r\leq n$ consider $MX=\mathsf0_n$ where $X\in\Bbb Z^{n\times 1}$ holds. We also have a $\mathsf{rank}(N)=m\leq n$ matrix $N\in\Bbb R^{m\times n}$ ...
6
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1answer
115 views

Algorithm that solves every Mixed Integer Linear Program (to optimality)?

Given a Mixed Integer Linear Program with rational coefficients (both for the objective functions and all constraints), is it always possible to solve it algorithmically? I know that you usually ...
2
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30 views

Pcross-like, nonogram-like in near-linear time [closed]

I have a problem with a puzzle game like pcross in which I have a nxn square: At any index of rows and columns I have an integer that say the maximum numbers of points that I can place in that ...
2
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1answer
85 views

Complexity of Deciding Feasibility of a system of linear inequalities over restricted variables

I am working out an interesting problem and would like some help with this particular sub problem: Suppose we have a matrix $ M =\left\lbrace a_{ij}\right\rbrace $ of size $n\times m$ where $ ...
0
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1answer
104 views

Convert general optimization problem to LP problem

I am trying to convert the following problem into a linear programming problem: There are $M\times N$ matrix $T$ of real numbers between 0 and 1 and $N\times 1$ vector $w$ of real numbers between 0 ...
0
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1answer
52 views

When to use non-negative-least square and least-square [closed]

What are the typical case we need to use Non-negative least squares NNLS $$ ||Ax - B||^2 $$ instead of least-square $$ Ax-B$$ (or vice versa)? And is there any drawback in applying them on large ...
2
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1answer
85 views

Better alternative to solve quadratic programming for large matrices

I have the following problem. Let's say we have $x_{jk}$ it is an expression value of gene $j$ in a sample $k$. It is the average of expression levels across the cell types $s_{ij}$, weighted by ...
3
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97 views

Testing if a point is inside a convex polytope formed by halfspaces in n-dimension

Assume we have a convex polytope that is formed by the intersection of $k$-halfspaces in $\mathbb{R}^{n}$. $$ a_{0,0}x^{n-1} + {a}_{0,1}x^{n-2} + ... a_{0,n-1} \leq 0 $$ $$ a_{1,0}x^{n-1} + ...
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1answer
59 views

Introducton books for ‎$\frak{E}_p(I)$

Are there any good books different from abstract harmonic analysis by hewitt to study ‎$\frak{E}_p(I)$. where ‎$\frak{E}_p(I)$ is: ‎Let $I$ be an arbitrary index set‎. ‎For each $i\in I$ let $H_i$ ...
4
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45 views

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 ...
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1answer
71 views

approximate diameter of polytopes in high dimensions

I just came across the following problem: Let us consider the unit corner of the n-cube $$ \Delta^n = \left\{(t_1,\cdots,t_n)\in\mathbb{R}^n\mid\sum_{i = 1}^{n}{t_i} \leq 1 \mbox{ and } t_i \ge 0 ...
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81 views

Listing all Lattice Points in a Box

Let $B := [-1,1]^n$ be an $n$-dimensional box. Moreover, let $v_1,\ldots,v_n \in \mathbb{R}^n$ form a basis of $\mathbb{R}^n$, where the entries of the $v_i$ are explicitly irrational. We can assume ...
3
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2answers
119 views

Has anyone developed a technique to generate a polytope given (possibly redundant) inequality constraints? [closed]

I've found a few papers that deal with removing redundant inequality constraints for linear programs, but I'm just trying to find the vertices for a feasible region, given a set of inequality ...
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66 views

Finding all feasible solutions

Let $u$ be a $n_{max} \times m$ matrix. Let $z$ be a $n_{max} \times s_{max} \times n_{max}$ cube. Let $w$ be a $n_{max} \times 1$ vector. All the three matrices can have values from the set $\{ 0, ...
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58 views

Finding optimal linear transformation for intersection of convex polytopes

I previously posted this on MathSE and am now trying here. I have the following situation, as shown in the following diagram: $W=\{w_i\}_{i=1..|W|}$ is a set of vertices (show in diagram in ...
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48 views

The column generation technique on a Train Unit Assignment Problem [Linear Programming]

I am doing an assignment where I need to implement a mathematical model that I can't wrap my head around. For the technique of column generation, one would need to my understanding, a master problem ...
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70 views

Is there a space in which the $\vec a$ in $\sin(a_1\cdot x)+\sin(a_2\cdot x)$ is linear?

Suppose one has equations of the form $\sin(a_1\cdot x_i)+\sin(a_2\cdot x_i)=y_i$ for $i = 1, \dots, n$ (there are also amplitudes and phase shifts, but let's ignore these for now). I want to solve ...
3
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1answer
345 views

Find the minimum distance between two convex hulls

We work over $\mathbb{R}^N$. Let $\mathbf{P}_1$ denote the hyperplane constructed using $N$ points, each of which is on a different axis (there are $N$ axes). We denote by $\mathbf{P}_2$ the convex ...
4
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1answer
141 views

Constrained optimization (QCLP) over $x$ with the constraint $x = Az$

I have a problem that looks very much like a (norm-constrained) linear program, but with an extra constraint that is unusual for me. The problem is, given a matrix $A$ and a vector $w$, $$ \min_{x ...
5
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1answer
113 views

Resource Constrained Routing with Refueling

What are good algorithms (resp. models) for calculating optimal or near optimal routes while taking into account fuel consumption, options for refueling and, limited tank capacity? Especially modeling ...
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47 views

Finding a movement taking out of a convex set

There is a convex set $S$ as the hull of M points in an D-dimensional Euclidean space and a point $\vec P$ in the set. Then, there is a set of vectors $\vec W$ taking the form $\vec W=\sum_{i=1}^N ...
4
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1answer
122 views

Basic result in semi-infinite linear programming

Consider a standard linear program of the form $$\textrm{minimize}_x~~~~ c^Tx~~~~ s.t. \\ Ax = b \\ x \geq 0$$ with $x\in \mathbb{R}^n$ and $A \in \mathbb{R}^{m \times n}$. It is well known that, if ...
2
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1answer
170 views

How can I find the maximum value of this function?

For given values of $A \in \mathbb{R}^{m \times n}, b \in \mathbb{R}^m$, how can I find the value of: $$ \max_{x \in [0,1]^n} \|Ax+b \|_1 $$ Or is this problem NP-hard?
2
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1answer
75 views

Algorithm to find the vertices of the equidistant lines between N closed polygonal lines

I have a set $\{C_1, C_2, \ldots, C_N\}$ of $N$ nonintersecting closed piecewise linear curves on the Euclidean plane. For every point $x \in \mathbb{R}^2$ we say it belongs to a territory serviced by ...
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64 views

Smallest sum of original column entries in 2d matrix [closed]

I have an interesting optimization problem I am trying to solve now and I thought I'd share it here in order to find the best answer. The problem itself is not complicated and it is stated like this: ...
3
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1answer
302 views

Constrained vs Unconstrained Optimization

I'm currently working on an optimization problem with a linear objective with linear and nonlinear constraints, i'm facing difficulties reaching a good solution, so i was advised to move the nonlinear ...
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1answer
197 views

Computing probability that $Ax\geq0$ where $x$ is a vector of iid gaussians and $A$ is matrix of $1$s and $0$s

This question came up in my research: What is the probability that $Ax\geq0$ where $x$ is a vector of iid gaussians and $A$ is matrix of $1$s and $0$s? So far I only figured out that I can do Monte ...
5
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68 views

Finding the optimal mixture of two convex functions

I am trying to find an efficient way to solve the problem $$\min_{p,x_1,x_2} p\cdot f(x_1)+ (1-p) \cdot f(x_2)~~~~~ s.t.\\p\cdot g_1(x_1) + (1-p)\cdot g_2(x_2)\leq 1 \\ 0\leq p \leq 1$$ where ...
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62 views

Derivation of gradient of SSE in Geodesic Regression

On page 79 (or page 5) of this this paper the gradient of the SSE of the Geodesic model is described explicitly. My question is how are these equitations derived in detail; where can I find the ...
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1answer
294 views

Computation of extreme rays of rational polyhedral cones - Hemmecke's project and lift algorithm

I am working on an implementation of Raymond Hemmecke's algorithm for finding generating sets of cones: http://arxiv.org/abs/math/0203105 Unfortunately I am struggling to make the algorithm work on ...
2
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0answers
66 views

Select n vectors from k vectors (in 3D) such that each component of the resultant vector >= each component of a given vector M

this was left unanswered for 1 week on MStackExchange, so I thought MOverflow would be more appropriate. Thanks :) Let $R = (R_x, R_y, R_z)$ be the resultant vector of the n vectors and $M = (M_x, ...
2
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91 views

How to solve the following generalized quadratic programming problem [closed]

I want to solve a generalized form of a quadratic programming problem $$\min_x \left(\sqrt{x^TPx}+\sqrt{x^TQx}\right)^2+c^Tx$$, $$\textrm{ s.t. } Ax\le b.$$ Here, $P$ and $Q$ are both positive ...
2
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1answer
136 views

Network flows with shared capacities

Suppose we have a flow network, with capacity constraints on weighted sums of arc flows, such as: $$2 f(1, 2) + 3 f(4, 5) + f(3, 7) \leq 10,$$ where $f(1, 2)$ denotes the flow through arc $(1, ...
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117 views

A specific case of the $p$-center problem

Given a fixed positive integer $m$, let $\cal{S}$ be the subset from $\mathbb{R}^m$ defined as $\cal{S} = \{u \in \mathbb{R}^m \mid \forall i \in \{1, \dots, m\}, u(i) > 0$ and $\sum_{i=1}^m{u(i) = ...
0
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1answer
85 views

accelerate convex optimization by proximal projection

I am using level method to solve non-smooth convex programming problem (where the objective function is given by an oracle from another program ): http://www2.isye.gatech.edu/~nemirovs/Lect_EMCO.pdf ...
5
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1answer
354 views

Examples of Polyhedra with Large Shadows

Let $P \subseteq \mathbb{R}^n$ be a polyhedron described by $\mathcal{O}(n^{c_1})$ inequalities, where $c_1$ is a constant. Moreover, let $M\colon P \to \mathbb{R}^2$ be a linear mapping. I'm looking ...
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1answer
124 views

An optimization problem in complex space

Consider the following optimization problem $$ \min \| \textbf{Ax-B}\| $$ $$ s.t.|x_i|=1,i=1,...,n $$ where $\textbf{x}\in \mathbb{C}^{n}$ is the optimization varaible, $x_i$ is the $i$-th ...