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1
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1answer
175 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
votes
0answers
47 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
30 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
votes
0answers
22 views

Explicit solution for a linear program with two constraints [migrated]

This is not a homework problem, although it wouldn't surprise me if it happens to exist in a textbook somewhere. Is there an explicit solution for the linear program $$\max_x c^Tx ~~ s.t. \\ d^Tx = q ...
1
vote
1answer
44 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 ...
1
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0answers
45 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
60 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
101 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
0answers
88 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
votes
1answer
48 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
votes
1answer
311 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 ...
1
vote
1answer
66 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 ...
5
votes
1answer
161 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
171 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 ...
2
votes
1answer
89 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{ ...
7
votes
2answers
191 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
93 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 ...
2
votes
1answer
136 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 ...
0
votes
1answer
267 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 ...
0
votes
1answer
155 views

Linear programming with exponentially many constraints and variables [closed]

From class, I learnt that problems like traveller salesman have a Linear programming representation with exponentially many constraints. Using method of separation, this problem is solved rather ...
1
vote
2answers
124 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 ...
0
votes
1answer
67 views

Generalized assignment problem with no integrality gap

Suppose I am solving the generalized assignment problem, so that I am given matrices $U$ and $W$ and a vector $c$ (all three of which have, say, positive entries), and I want to solve ...
1
vote
1answer
78 views

ILP for minimum edge coloring problem

We know that for a Graph G=(V,E) ,minimum edge coloring is a coloring of E, i.e., a partition of E into disjoint sets E1,E2...,Ek such that, for 1<=i<=k, no two edges in Ei share a common ...
2
votes
1answer
162 views

Multiplicative gradient descent?

The normal gradient descent is additive: $w_{t+1}=w_t-\lambda_t\nabla f(w_t)$, but is there a multiplicative gradient descent that looks something like $w_{t+1}=w_t[-\lambda_t\nabla f(w_t)]$? I know ...
6
votes
1answer
176 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 ...
0
votes
0answers
26 views

Find relationships between events

I have a set of Events $(E_i)_i$ which have a probability $(P_i)_i$. I am able to write each event as a sum of distinct events that form a partition of the space. My goal is to find all the ...
2
votes
1answer
76 views

Find base of kernel with as many 0 as possible

I have a 400x132 rectangular matrix with only 0 and 1. I am looking for the linear combinations of the columns of the matrix that sum to 0. For example C1 + C2 - C3 = 0. I want to find the linear ...
2
votes
2answers
114 views

Boundedness of ratio of linear functions

Consider the function \begin{eqnarray} f(x_1,x_2,\cdots, x_n) = \frac{\sum_{i}^{n}a_ix_i}{\sum_{i}^{n}b_ix_i}, \end{eqnarray} over the set $S = \{x := (x_1,x_2,\cdots, x_n):-1 \leq x_i \leq 1,\; ...
0
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0answers
67 views

polynomial 0,1 integer programming

IS there a mathematical optimization branch that explicitly tries to optimize this (type) problem? $$\eqalign{ & \min \cr & \sum\limits_{i = 1}^N {(J*s[i] + {J_1}*s[i]*s[i + 1] + ...
0
votes
1answer
85 views

Solution of infinite dimension linear system

Suppose that ${a_n}$ and $b_n$ is decreasing sequence such that $a_0=A$, $lim_{n->\infty}a_n=0$ and $b_0=B$, $lim_{n->\infty}b_n=0$. For fix n, we can construct n dimension linear equation ...
2
votes
2answers
250 views

convex polytope integer points

is there a simple proof for the following lemma: An unbounded convex polytope (defined by linear constraints) has either zero integer points or infinite many integer points.
1
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0answers
124 views

Reduce a Combinatorial problem

It is given n sets with k vectors. (k is element-wise positive or zero) Choose one vector of each set so that the biggest element of the sum of the chosen vectors is minimal. What i also know but is ...
0
votes
2answers
118 views

Approximate solution to large mixed integer programming problem

What are the available approaches to find an approximate solution to a large mixed integer programming problem? I ran my problem in the Gurobi MIP solver. It can find a feasible solution in ...
1
vote
1answer
123 views

a closed form lower bound solution for linear programming

Given a linear objective function and a system of linear constraints, is there any known closed form lower bounds for it? to clearly express the problem assume that $$ z(\mathbf{a,B,c})=\mathop {\inf} ...
2
votes
1answer
232 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 ...
2
votes
1answer
150 views

Optimization problem whose cardinality never exceeds 7 for some reason

I am working on a problem in which I have a collection of $n$ points, $x_1,\dots,x_n$, in the plane, as well as a positive definite matrix $\Sigma$ and another point $\mu$ in the plane. I am trying ...
0
votes
1answer
36 views

Large scale least squares of non symmetric and non square problems

Given a system like $b=Ax$ with an non symmetric and non square $A$ I would like to solve it having many elements in $x$ (lets say $10^7$). There is a large amount of algorithms for symmetric ...
1
vote
1answer
54 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 ...
1
vote
1answer
143 views

Calculate the discrete set of points B which are in the convex hull of the set of points A

This problem is likely best described with the following picture: Given the discrete set of points $A$ (shown in blue), I wish to calculate the discrete set of points that are contained within the ...
2
votes
2answers
110 views

sensitivity analysis in conic optimization

I have a conic optimization of the form: $\min_x \langle c, x \rangle$, s.t. $Ax = b$, $x \in K$. Where $x \in \mathbb{R}^{n}$, $A$ is an $m \times n$ matrix, $b \in \mathbb{R}^m$, $K$ is a self ...
2
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0answers
65 views

Algorithms to find the solutions of a homogenous matrix equations for non-commutative rings

In one paper from 1980 I found a note that there are no known algorithms for solving homogenous matrix equations $x \cdot M = 0$ for matrices which elements belong to a non-commutative ring. (The ...
1
vote
2answers
54 views

Linear Programm with matrix [closed]

Is there a name for problems like this min norm(Cx) Ax = b where C is a matrix and norm is the maximum norm. This is kind of like a linear Programm. Could this be rewritten as linear programm? Or Any ...
0
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0answers
82 views

Inverse problem with a rank-1 update

I hope you can help me out with this. I have to find the solution x to an inverse system $$ x=A^{-1}b $$ This inverse problem is basically a least square problem with a rank-1 update. $$ ...
3
votes
1answer
127 views

submatrix of a given size with maximum frobenius norm

Let $I\subset \{1,2,\ldots,n\}$, and let $|I|$ denote its cardinality. Now given a Hermitian matrix $\mathbf{A}\in\mathbf{C}^{n\times n}$. I am interested in finding the subset $I$ that maximizes the ...
1
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0answers
62 views

integrality of a linear program — binary equality constaints

Consider the following linear program: $\left\{ \begin{array}{l} \underset{x}{max} \;\;c^Tx\\ [I, \;B]x = \mathbf{1}\\ x\geq 0 \end{array} \right.$ where $c$ is a vector ...
1
vote
1answer
165 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 ...
22
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2answers
867 views

An Interesting Optimization Problem

You are given n non-negative integers $a_1, a_2 ,, a_n$. In a single operation, you take any two integers out of these integers and replace them with a new integer having value equal to difference ...
4
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2answers
133 views

combinatorial and linear duality

Let $S$ be a finite set, and let $W$ be a nonempty set of subsets of $S$; we will identify every subset of $S$ with its characteristic function, a 0-1 vector in $\mathbb R^S$. The combinatorial dual ...
2
votes
1answer
61 views

Integer point in a non-empty polytope

I have a high-dimensional, non-empty polytope $Ax\geq b$ sitting inside the cube ($0\leq x_i \leq 1$). Is there any general theory or technique to show that this polytope contains an integer point, ...
1
vote
0answers
62 views

Are there any known bounds on the value of solutions of linear integer programming?

Given a linear objective function and a system of linear constraints; are there any known bounds on the values of (positive) integral solutions in terms of the coefficient matrix of the constraints? ...