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1
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1answer
270 views

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 ...
-1
votes
0answers
42 views

About perturbation of spectral radius of a matrix because of diagonal perturbations

Say I have an off-diagonal symmetric $0,1,-1$ matrix $B$ and a set of $2k$ diagonal matrices, $D_{11}, D_{12}, D_{21}, D_{22},..,D_{k1},D_{k2}$. (you can assume the diagonal matrices to be such that ...
2
votes
0answers
57 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 ...
-2
votes
0answers
36 views

Global minimization. How? [closed]

I know it's impossible to have an algorithm that finds the global minimum (without a brute force approach), for a general problem. I also understand that the efficacy of the flavour of minimization ...
3
votes
2answers
80 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 ...
1
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0answers
53 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, ...
1
vote
0answers
49 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 ...
1
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0answers
34 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 ...
3
votes
1answer
328 views

Name search for special Linear Integer Program

I am looking for a name for the following question in literature! (and if you know it, then it would be great) I couldn't find it and due to wide audience here, presumably you know more. Thank you ...
0
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0answers
55 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 $i=1...n$ (there are also amplitudes and phase shifts, but let's ignore those for now). I want to solve for $\vec ...
0
votes
1answer
89 views

Is it possible to represent non-linear ranking type constraints as equivalent linear constraints?

I have formulated a linear program with binary indicator variables $z_i(a)$ which is equal to $1$ if the $i^{th}$ document is of rank $a$ and $0$ otherwise. The other variables in the linear ...
1
vote
1answer
54 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 ...
3
votes
1answer
117 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 ...
5
votes
1answer
85 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 ...
2
votes
1answer
68 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 ...
0
votes
0answers
44 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 ...
2
votes
1answer
161 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?
0
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0answers
15 views

How to solve a special linear system with noise?

Sorry the title may be confusing. I'm not so sure how to categorize this problem. Anyway, We have a real-valued vector $\overrightarrow a=(a_0,a_1,...,a_{2^n-1})$. Each $a_i$ comes from the inner ...
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votes
3answers
3k views

Solving a system of linear inequalities — what is the dimension of the solution set?

It is well known how to solve a system of linear equations $A{\bf x} = {\bf b}$, but how do we solve a system of linear inequalities $A{\bf x} \leq {\bf b}$? For the applications I have in mind the ...
2
votes
1answer
42 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 ...
1
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0answers
62 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
votes
1answer
237 views

How to implement linear constraints that include several absolute values

Dear all, I am trying to implement a linear constraint that includes several absolute values in the form: Abs(A) + Abs(B) + Abs(C) + Abs(D) + ... = 1 Since the minimization problem includes quite a ...
1
vote
1answer
115 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 ...
3
votes
1answer
111 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 ...
4
votes
3answers
565 views

Algorithm for the intersection of a vector subspace with a cone of non-negative vectors

Hi, I would like to know whether there is some more effective way of how to compute an intersection of a vector subspace of $\mathbb{R}^{n}$ with a cone of vectors with non-negative entries than the ...
1
vote
1answer
191 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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0answers
61 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 ...
2
votes
0answers
45 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 ...
1
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0answers
51 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
votes
0answers
76 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
votes
1answer
114 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, ...
6
votes
1answer
212 views

Speed up Linear programming

I have a linear programming problem like this: minimize $c^t X$ under the constraint that $AX \ge b$. I will need to solve this linear programming problem online many times. I need it to be as fast ...
6
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0answers
98 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) = ...
2
votes
1answer
98 views

Fastest 'Oracle' Algorithm for satisfying a single linear constraint on a convex set?

In this paper by Arora, Hazan, and Kale (http://www.cs.princeton.edu/~arora/pubs/MWsurvey.pdf) a method is given for using the Multiplicative Weights Update algorithm to quickly solve Linear Programs ...
5
votes
1answer
331 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 ...
0
votes
1answer
61 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 ...
3
votes
3answers
3k views

Maximum flow with negative capacities?

I'm trying to compute an (s-t) maximum flow through a network which includes a number of arc pairs ((u,v), (v,u)) that have equal, negative capacities (weights). I'm not aware of any efficient ...
1
vote
1answer
1k views

Maximizing linear objective function with absolute values

This has be asked on other forums, though couldn't find authoritative answer. I have a linear program over the reals and don't want to introduce integer or binary variables. The objective function ...
0
votes
1answer
658 views

eigen-decomposition solution? is it unique?

Assume an N*N covariance matrix (Q) which is a positive definite matrix. The decoder X is assumed to be N*s, where s<=N. X is calculated to be s eigenvectors corresponding to s minimum eigenvalues. ...
5
votes
1answer
224 views

Algorithm to minimally connect line segments in Euclidean plane

Suppose you have finitely many line segments in the Euclidean plane. How do you "connect them to form one chain of line segments of minimal length?" More formally and generally, what I'm looking for ...
7
votes
1answer
177 views

A polytope with a bound on the sum of any $k$ variables

Let $2\le k\le n-1$ and define the polytope $$P_k(n) = \lbrace (x_1,\ldots,x_n) \in \mathbb{R}^n : -1\le x_{i_1}+\cdots +x_{i_k} \le 1 \text{ for all } 1\le i_1\lt\cdots\lt i_k\le n\rbrace.$$ There ...
1
vote
1answer
83 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 ...
2
votes
1answer
91 views

Standard names and methods for this type of fitting minimization

In material science research, we have come across the following type of problem. Given a m by n matrix A, a m vector b, and error tolerance $\varepsilon$, we want to do this minimization $$\eqalign{ ...
2
votes
1answer
143 views

books on very large scale linear optimization

Recently in my material science research, I have encountered problems of very large scale linear optimization. I read the introductory book "Introduction to Linear Optimization (Athena Scientific ...
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2answers
194 views

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 - ...
0
votes
2answers
99 views

Inner Product of Given Sum Positive Sequence

Let $$A = \Big\{(a_1,a_2,\dots)\ \Big|\ a_i\ge 0, \sum_{i=1}^\infty a_i=1\Big\},$$ $$v(x)=\sup\left(\bigg\{\sum_{i=1}^\infty a_ib_i\ \bigg|\ (a_i)_{i=1}^\infty,\, (b_i)_{i=1}^\infty \in ...
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3answers
686 views

Deciding membership in a convex hull

Problem: Given points $u,v_1,\dots,v_n\in\mathbb{R}^m$, decide if $u$ is contained in the convex hull of $v_1,\dots,v_n.$ This can be done efficiently by linear programming (time polynomial in ...
1
vote
1answer
56 views

Heuristic for choosing n-vectors from n-sets

my given problem is: choose n-vectors from n-sets (one vector from each set) so that the biggest element in the sum of the chosen vectors is minimal. Unfortunately the problem is NP-hard. So I'm ...
0
votes
1answer
599 views

Convert linear programming problem into its standard form [closed]

all, I met a question that, the cost function of the linear programming problem is a function with absolute value. Here is the problem: min 3x1+|6x2+3| st. |x1+4|+|2x2|<=3 How can I deal with ...
1
vote
2answers
126 views

investigating positivity/negativity of a function [closed]

I'm investigating if and how the positivity or negativity of a multivariable function can be proved. Consider $y_{1},y_{2},y_{3}\in\mathbb{R}$ and the following function ...