All Questions
685 questions
2
votes
1
answer
1k
views
Finding integer points inside of a parallelogram
Suppose $P = \{p_1,\ldots,p_4\} \in \mathbb{R}^2$ defines a quadrilateral (here, specifically, a parallelogram). In the particular case I'm dealing with, I know that there exists at least one point ...
- 1,318
1
vote
0
answers
256
views
Equal maximum and minimum in a large-scale linear programming
For a linear optimization of an integral (with integral constraints), I perform a linear programming for the equivalent series. Maximum and minimum of the LP problem tend to be equal as I increase the ...
- 111
2
votes
0
answers
39
views
In what paper was the shrinkage parameter introduced to the nelder-mead simplex direct search algorithm?
I have read lots of papers referencing a 4th shrinkage parameter when talking about the Nelder Mead Simplex method. However, I cannot see any shrinkage parameter in the flow chart of the original ...
- 21
2
votes
0
answers
163
views
existence of lattice point in polytope
This question was probably asked before but here goes. I have a convex polytope given by $Ax\leq b$ for a specific integer matrix $A$ and integer vector $b$. I need a simple method/result on how to ...
- 501
2
votes
1
answer
689
views
Why does the LP Formulation of the MST Problem need Topology Constraints?
I am looking for an example that demonstrates the necessity of either subtour-elimination or of connectivity constraints in the LP formulation of the MST
In the internet I only could find the LP ...
- 13.2k
0
votes
2
answers
244
views
Rewrite optimization objective
Hi,
I wanted to ask, under which conditions can one rewrite the optimization objective
$\min_x f(x)\;\;\;s.t.\;\;\;g(x) \leq s$
as
$\min_x g(x)\;\;\;s.t.\;\;\;f(x) \leq t$
I have particular ...
- 409
1
vote
2
answers
306
views
Name of operations on two vectors
Suppose we have two vectors $x\in \mathbb{R}^n$ and $y\in \mathbb{R}^m$.
I could define the mapping
$$
T: \mathbb{R}^n\times \mathbb{R}^m \rightarrow \mathbb{R}^{n\times m}
$$
as follows
$$
T(x,y) = ( ...
0
votes
2
answers
340
views
positive semidefiniteness: a psd matrix substracted by another rank 1 psd matrix
Given that $A$ is a positive semidefinite matrix, $x$ is a vector, $\lambda_0 \in [0, +\infty) $ is a real non-negative number. I want to know the answer to the following optimization problem.
$$
\...
- 101
2
votes
2
answers
129
views
LP constraint enconding
I have an objective function to be maximized
$obj(x) = \sum_i \gamma_i x_i$ with $x_i \in \mathbb{R}$
With multiple constraints of the form:
$\min_{y \in 0,1} (\sum_{i \in A} \alpha_i x_i + \sum_{i ...
- 21
1
vote
1
answer
296
views
Deducing Linear Inequalities
Let $X_1,X_2,\ldots,X_n $ be indeterminates. Denote by $S$ the set of all linear inequalities of the form
$X_{i_1}+X_{i_2}+\ldots+X_{i_k} \geq k,$
with $k \in \{ 1,2,\ldots,n \}$ and $1 \leq i_1< ...
- 185
2
votes
1
answer
227
views
Arrangements of hyperplanes
Fix $n>0$ and $X\subseteq\mathbb{R}^n$. A function $f:X\longrightarrow\mathbb{R}$ is linear if it is of the form
$$
f(\bar{x})=a_1x_1+\ldots+a_nx_n+b
$$
for some $a_i,b\in\mathbb{R}$.
Suppose we ...
- 3,274
1
vote
1
answer
4k
views
What does "Vertex Solution" mean?
Hello!
I come across the word "vertex solution" in the context
" We can also assume that x and y are vertex solutions,so that the sequence {x,y} remains in a finite set."
Could anybody know any ...
- 13
4
votes
1
answer
396
views
Optimization problem - maximizing number of satisfied linear inequalities subject to a quadratic constraint
I am wondering what is known about optimization problems of the following type.
Our control x is a unit vector in $\mathbb{R}^n$. We are given a finite number of linear inequalities
$$Az≥b,$$
and we ...
- 185
0
votes
0
answers
104
views
Big eigenvalues of a special stochastic matrix
Given a matrix $M$ of size $n\times n,$ we write its different eigenvalues by $x_1,x_2,\ldots,x_m$ with $m\leq n$ such that $|x_1|>|x_2|>|x_3|>\cdots|x_m|,$ and call $x_2\doteq |\lambda_2|(M)....
- 105
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 + ...
- 123
9
votes
1
answer
3k
views
Inverse of a totally unimodular matrix
A unimodular matrix $M$ is a square integer matrix having determinant $+1$ or $−1$.
A totally unimodular matrix (TU matrix) is a matrix for which every square non-singular submatrix is unimodular. A ...
- 93
5
votes
1
answer
3k
views
Maximizing supermodular functions
I have a real supermodular objective function which I want to maximize with constraint. The constraint is on the size, like |A|=k .
I am wondering if anyone can give me more information about a ...
0
votes
0
answers
145
views
The role of subgradient in programming with nonsmooth functions
It is obvious that there is similarity between subgradient and gradient. The subgradient of smooth functions is reduced to gradient. I have two questions.
The first is does there exist subgradient ...
- 173
4
votes
1
answer
2k
views
lipschitz constant of a multivariate function
I have a function $f:\mathbb{R}^{50} \rightarrow \mathbb{R}$ and I need to compute the Lipschitz constant of $f$ to solve an optimization problem using a specific algorithm. Does any one have ...
- 41
2
votes
0
answers
63
views
Put positive polynomial in finite intersection of half-spaces
This is a cross-posting of a MSE question (which did not attract any attention there so far).
Denote by $V={\mathcal P}_{n,d}$ the space of polynomials in $n$ variables with degree at most $d$, ...
- 621
2
votes
1
answer
130
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 ...
- 121
2
votes
3
answers
752
views
Reference Request for Integer factorization with LP/ILP
Have there been attempts to factor integers with Linear Programming?
Searching the internet suggests that for Integer Factorization only Number Theoretic algorithms, like sieves, are taken into ...
- 13.2k
0
votes
1
answer
173
views
Avoiding epsilon in mixed integer linear and quadratically constrained programs
I would like to represent the following constraint as MILP constraint where $x \in [a, b]$ with fixed $a, b \in \mathbb{R}$ and $y \in \lbrace 0, 1 \rbrace$.
$(x = 0 \wedge y = 1) \vee (x \neq 0 \...
- 1
7
votes
1
answer
422
views
Generalization of the equilateral triangle?
I consider points in the two-dimensional plane.
An equilateral triangle is a set of three points in the plane which are equidistant.
Suppose now I have $n$ points $x_1,...,x_n$. What is the ...
- 840
1
vote
1
answer
360
views
for what arguments the function reaches maximum?
Hi,
What is the maximum of the following function?:
$f(x_i,w_i)=\frac { \sum w_i}{ \sum \frac {w_i}{x_i} } - \frac{ 1 - \prod \left ( 1 - w_{i}\right )}{ 1 - \prod \left ( 1 - \frac{w_{i}}{ x_i}\...
- 11
1
vote
1
answer
220
views
What kind of optimization problem is this?
I come across an optimization problem of the following form.
$$\max {\frac{1+v}{1-u}} \qquad \text{s.t.} \qquad ux^2+vy^2-xy \ge 0, \quad \forall x,y\in\mathbb{R}$$
I do not know much of optimization. ...
- 13
2
votes
1
answer
83
views
Computing a point of refraction
Oddball question: say I want to travel from $(a, b)$ where $b > 0$ to $(c, d)$ where $d < 0$ using the shortest path, where I can travel at velocity $v_1$ in the upper half-plane and at velocity ...
- 285
1
vote
0
answers
102
views
Trying to get an idea of the maths I could use for this optimization problem
Firstly, apologies if some of the notation or terminology is odd, or if I am defining functions that have standard notation associated with them already - I am not familiar with the concepts in this ...
- 11
2
votes
1
answer
2k
views
Finding a point farthest away from $k$ points in a polygon
There are $k$ points placed inside a polygon and I am interested in finding a point inside the polygon (not necessarily on its boundary) who's minimum distance to any of the $k$ points is maximized.
...
- 57
1
vote
2
answers
1k
views
Efficient algorithm finding 'a' solution of system of linear inequalities
I'm working on rational number field $\mathbb{Q}$.
Is there an efficient algorithm finding a solution of system of linear inequalities?
In many computer algebra systems like Sage or Maple,
there ...
- 627
3
votes
2
answers
791
views
complexity of finding optimal matchings of given fixed size
It is known, that maximal matchings (i.e. matchings with the maximal number of edges) and optimal matchings (i.e. matchings for which the sum of edge weights is optimal) can be calculated in ...
- 13.2k
1
vote
0
answers
1k
views
Robust optimization in matlab using fmincon [closed]
I am trying to implement the following optimization (from this paper) in Matlab using fmincon:
$\min_\omega\omega'\Sigma\omega$ subject to $\min_Ur_p \geq r_0$
where $\Sigma$ is a positive definite ...
0
votes
1
answer
2k
views
Find edge weights that fit given node weights
Let $G = (V,E)$ be a connected simple graph (unweighted, undirected, no selfloops) on $n$ nodes.
Let $\mathbf{d} := (d_1, d_2, ..., d_n) \in \mathbb{R}_{>0}^n$ be a vector of arbitrary given node ...
- 340
3
votes
1
answer
4k
views
Schur complement and negative definite matrices
Hello,
My question regards to the Schur complement lemma. Consider the matrix $M=\left( \begin{array}{cc}
A & B\\\
B^T & C \end{array}\right)
$.
According to the lemma $M\geq0$ iff $C>0$ ...
- 155
19
votes
2
answers
2k
views
Is the tensor product of polyhedra a polyhedron?
Conventions: A polytope in a finite-dimensional $\mathbb R$-vector space $V$ is defined to be a convex hull of finitely many points in $V$. A polyhedron in a finite-dimensional $\mathbb R$-vector ...
- 33.8k
1
vote
0
answers
126
views
Matrix Minimax problem
I have the equation $\Sigma_k(M_k{p_k})V=EV$, where the $M_k$ are n*n real Hermitian matrices, $V$ is a n*n eigenvector matrix, $E$ a dim-n energy eigenvector and the $p_k$ scalar parameters. The $M_k$...
- 4,829
4
votes
1
answer
585
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 ...
- 41
0
votes
1
answer
183
views
necessary and sufficient conditions for a function to be DC
Hi, Does anyone know the necessary and sufficient conditions for a function to be a DC-function?
Definition: A function is a DC-function if and only if it can be written as a differnece of 2 convex ...
- 1
0
votes
1
answer
1k
views
Finding the lowest cost set of disjoint paths using all nodes in a directed graph?
I have a directed graph with edges connecting nodes representing costs.
I wish to find the set of paths which
-go from node 'start' to node 'end'
-are node-disjoint (except for the start and end ...
- 3
0
votes
1
answer
2k
views
Finding linearly independent columns of a large sparse rectangular matrix
I have a problem that necessitates solving a large non-negative least-squares
problem. My matrix A is large, sparse, highly rectangular (num rows >> num cols)
and nearly binary. However, A is not ...
- 103
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 ...
- 191
5
votes
2
answers
888
views
relation between solution of a linear program and its perturbation
I have a linear program over a finite set of points $(x_1, x_2,\ldots, x_m)\in\mathbb{R}^n$:
$$
\max_j c' x_j
$$
Suppose the solution of this LP is obtained at a point $x_{j_1}$, which is a vertex ...
- 373
3
votes
0
answers
713
views
Minimum weight bipartite graph clique covering
I was wondering if anyone here could give me any pointers as to how to solve the following problem.
Let $B=(L,R,E)$ be a bipartite graph, and $\forall u\in L\cup R$, let $c_u$ be a cost associated to ...
- 151
1
vote
2
answers
1k
views
Nonstandard Hessian approximations in Gauss-Newton
The Gauss-Newton algorithm optimizes functions
$$
E(x) = \sum f(x)^2
$$
by approximating f as (locally) linear, in which case the Hessian of $E$ is approximated as
$$
H = 2 \sum {J_f}^T J_f
$$
Now ...
- 561
5
votes
2
answers
597
views
Gandhi's quote formalized [closed]
Hello,
I hope this question is appropriate for Mathoverflow. Gandhi said, "Be the change that you wish to see in the world". I don't understand anything in Game/optimization theory (I don't know ...
- 5,562
0
votes
1
answer
179
views
Is these two optimization problems share the same solution?
Hello all,
I am dealing with some SDP optimization, and I come across the following problem.
The optimization problem is given by
\begin{aligned}
&\operatorname*{min}_{t_1,\ldots,t_m,X}\ \sum ...
- 43
1
vote
2
answers
444
views
Levenberg-Marquadt near the minima for non-zero-residual problems
I'm using the LM algorithm to do gradient descent in a model fitting context. I'm minimizing:
$$
c(x) = \sum ( f_i(x) - y_i )^2
$$
I'm noticing that after a few steps when I'm close to the minima, I ...
- 561
2
votes
1
answer
126
views
Is it possible to represent non-linear ranking type constraints as equivalent linear constraints?
I have formulated a linear program with binary indicator variables $z_i(a)$ which is equal to $1$ if the $i^{th}$ document is of rank $a$ and $0$ otherwise.
The other variables in the linear program,...
- 121
0
votes
0
answers
103
views
Gauss-Newton for quotient functions
I'm optimizing a function of the form
$$
\sum \frac{ \|\mathbf{f_i}(x)\|^2 }{ g_i(x)^2 + h_i(x)^2 }
$$
where $x$ is a real vector, $\mathbf{f}(x)$ is a real vector, and $g(x)$ is a scalar. My first ...
- 561
1
vote
2
answers
524
views
non convex optimization
Hi there,
In my studies I come up with this nonconvex optimization problem
argmin |Ax|_2+lamda*|x|_1 subject to x'x=1
where cost function is nonsmooth but convex and the constrant in nonconvex.
I ...
- 13