All Questions
686 questions
3
votes
2
answers
2k
views
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
views
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
0
votes
1
answer
150
views
Differences between the convex discrete maximization and minimization problems? [closed]
Would you tell me some main distinctions between the convex discrete minimization and maximization optimization problems?
In the case of the feasible are bounded then we only need to transform one ...
- 54.2k
1
vote
0
answers
187
views
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
1
vote
0
answers
83
views
What is optimal distance between inverse of convolution operator?
I am looking for a measure to find the optimal distance measure between inverse of an convolution operator $A$ and say another convolution operator $B$. I want my measure to be sharp that mean when $B$...
2
votes
1
answer
171
views
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
vote
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
0
answers
232
views
Semi-convex problem and almost convex problem
I have a target function, I've computed its Hessian to check convexity, it has a positive-definite sub-matrix and small negative-definite sub-matrix and a kernel. Sometimes it is even better -- the ...
- 355
1
vote
1
answer
1k
views
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
1
vote
0
answers
94
views
About a particular definition of a Hessian of a function of tuples of matrices
Say I have a function $L : (W_1,..,W_{H+1}) \rightarrow \mathbb{R}$ i.e it takes a tuple of $n$ matrices of different dimensions and computes a number from them.
Then I see being defined a ...
- 2,246
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
2
votes
0
answers
234
views
Characterizing the optimimum over the space of probability measures
Consider the following optimization problem:
\begin{equation}
\max_{\mu \in \mathcal{M}} \int \log\left( \int e^{\alpha U(x,y)} d\mu(y) \right) d\nu(x)
\end{equation}
where $\mathcal{M}$ is the space ...
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
0
votes
0
answers
243
views
Limit of argmin of sum
Suppose that I know $f_n\rightarrow f$ and $g_n\rightarrow g$ are both continuous maps from a Complete Riemmanian Manifold $X$ to $\mathbb{R}$ which converge pointwise almost everywhere. Then is it ...
- 5,405
1
vote
0
answers
55
views
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
2
votes
0
answers
71
views
Existence of probability distribution satisfying upper/lower bounds on events
Suppose we have a finite sample space $S$ and some events $A_1, \dots, A_k \subseteq S$. We would like to put a probability distribution on $S$ so that no element has probability greater than a ...
- 21
5
votes
2
answers
2k
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 ...
- 103
2
votes
1
answer
201
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 ...
- 32.5k
8
votes
0
answers
1k
views
Infinite Linear Programming
I'm trying to prove optimality for a continuous linear program. That is, I have a linear program with an uncountable number of variables and constraints. I'm not sure how to demonstrate feasibility ...
CommunityBot
- 1
3
votes
2
answers
313
views
Uniqueness of fixed points for rational transformations
Background
Let $a,b,c,d$ be nonnegative constants and consider the map $T\colon [0,1]\times[0,1] \rightarrow [0,1]\times[0,1]$ defined by
$$
T(x,y) := \left( \frac{1}{1 + ax + by}, \frac{1}{1 + cx + ...
- 19.3k
1
vote
1
answer
241
views
Covering max flow arcs by arc disjoint paths
Let $(N,A,s,t,u)$ be a network with node set $N$, arc set $A$, source $s\in N$, sink $t\in N$ and capacity vector $u\in\{1,2,\ldots,T\}^A$, and let $x=(x_a)_{a\in A}$ be a maximum $(s,t)$-flow. Is it ...
- 19.3k
7
votes
0
answers
1k
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 & \...
2
votes
1
answer
126
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 program,...
CommunityBot
- 1
0
votes
1
answer
543
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 ...
CommunityBot
- 1
1
vote
0
answers
90
views
Separable Least squares - is there a notion of conjugate directions?
I have a general question.
Suppose I have the following to optimize
$$\|Y-A(\mathbf{x})B(\mathbf{y})\|^2$$
where $Y$ is a vector, $A(\mathbf{x})$ is a matrix that depends on a vector $\mathbf{x}$ in a ...
- 61
5
votes
1
answer
725
views
Optimization of a function of a positive definite matrix and its inverse
This question is a little ill-posed, but I've been playing with some equations and am just wondering if this resembles any known problems that have been solved.
Suppose I have two real, positive ...
- 570
0
votes
1
answer
204
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 ...
- 19.6k
6
votes
6
answers
3k
views
Circumference of Convex Shapes
Here is a puzzle I found in Mitteilungen der DMV (roughly, "Letters of the German Society of Mathematicians"), issue 19/2011. It was posed by Alfred Schreiber in "Wie man Hasen fangt" (How to catch ...
CommunityBot
- 1
1
vote
1
answer
391
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 ...
- 3,230
2
votes
1
answer
682
views
Maximal minimum for a sum of two (or more) cosines
Please prove (or disprove, and give the correct answer):
$$2 =\mathrm{argmax}_{r\geq 1}\min_{x\in \mathbb{R}}\left[\cos\left(x\right)+\cos\left(rx\right)\right]
$$
In other words, find $r \geq 1$, ...
CommunityBot
- 1
-1
votes
2
answers
775
views
Is finding a local minimizer of a NP-hard optimization problem is still NP-hard [closed]
I was wondering if for a NP-hard optimization problem, I only want to find its local minimizer, is it still NP-hard or NP-hard is only true when trying to find a global minimizer?
- 11
1
vote
0
answers
36
views
Deterministic global solution to find the Optimal-knot placements for continuous piecewise linear functions to fit nonlinear data
I have been searching lately for a deterministic global technique to linearize a nonlinear function with continuous piecewise linear regions.
I've a univariate nonlinear function y=f(x). where f(x) ...
- 11
2
votes
0
answers
177
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 ...
- 21
1
vote
0
answers
88
views
solution of an infinite horizon optimization problem
Give the following formulation:
$\min_{\{x_s(t):\forall s,t\}} \sum_{s \in \mathcal{S}} \mathbf{1}\left(\lim_{T\rightarrow \infty} \frac{1}{T} \sum_{t=1}^T \frac{y_s(t)}{x_s(t)}\leq 1\right)$
$s.t. ...
1
vote
3
answers
2k
views
How to solve Linear Programming problem with tighter Integer Programming constraints
I want to learn a bit about Linear Programming.
After some research, I decided to solve the Cutting Stock problem as an example to learn. After doing some more research, I feel like I finally ...
CommunityBot
- 1
6
votes
1
answer
263
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 ...
- 2,464
2
votes
0
answers
100
views
Solve non-linear Optimization Problem [closed]
I have to find $x$ that minimizes: $$ \sum_{k}(x^H\textbf A_kx - b_k)^2$$ where $A_k$ are 4 x 4 positive definite matrices ($A_1, A_2,...A_k$), $x$ is 4 x 1 vector and $b_k$ are scalars ($b_1,b_2,......
- 121
2
votes
0
answers
112
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 row/col....
2
votes
1
answer
531
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 $ a_{ij}\...
- 25.4k
0
votes
1
answer
2k
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 $A$...
- 188k
3
votes
1
answer
260
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 ...
- 12.7k
5
votes
1
answer
106
views
Two fold optimization: is there an established approach for this kind of problem?
I have a list of thousands of linear expressions, with less than 100 total variables. Each expression is associated with a positive point value. I want to make as many of the expressions greater than ...
- 2,966
7
votes
0
answers
2k
views
Optimization over space of probability measures
Consider an optimization problem as follows:
$$
\min\mathbb E_w[f_0(w)] \mathrm{\,\,\,\,\,\ s.t.\,\,\,\,} E_w[f_i(w)]\leq 0 ,\,\,\, i=1,\dots, k
$$
where the maximum is taken over $\mathscr M$, ...
CommunityBot
- 1
2
votes
0
answers
71
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, ...
- 21
3
votes
2
answers
2k
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 $E_1, E_2, \dots, E_k$ such
that, for $1 \leq i \leq k$, no two edges in $...
- 2,339
4
votes
0
answers
405
views
maximize non-convex composite function
I want to maximize a composite function over a convex set
\begin{equation}
\begin{aligned}
& \underset{\mathbf{p}}{\text{maximize}}
& & f(\mathbf{p})-g(\mathbf{p})\\
& \text{subject to}...
- 61
3
votes
0
answers
970
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} + {a}_{1,...
- 31
0
votes
1
answer
82
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 $H_i$ ...
- 11