The linear-programming tag has no usage guidance.

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814 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 ...

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74 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 ...

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**1**answer

786 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 ...

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votes

**1**answer

530 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 ...

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44 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 $d$, ...

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votes

**1**answer

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 ...

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vote

**1**answer

378 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 ...

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vote

**1**answer

390 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.
...

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**2**answers

635 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 ...

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vote

**2**answers

129 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 ...

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vote

**0**answers

721 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 ...

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**1**answer

527 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 ...

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vote

**1**answer

1k 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 $C>0$ ...

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**2**answers

824 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 ...

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104 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 ...

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**1**answer

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 ...

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**1**answer

743 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 ...

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votes

**2**answers

390 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 ...

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**2**answers

319 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 ...

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**2**answers

552 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 ...

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**2**answers

163 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 ...

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**1**answer

90 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 ...

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93 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 ...

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**1**answer

297 views

### Results for minimizing the norm w.r.t a unitary matrix

Suppose $x \in \mathbb{R}^n$, $B,U \in \mathbb{R}^n\times\mathbb{R}^n$ and $U$ a unitary matrix. Define $g_{U}(x) = || BUx||$ where $||.||$ is some norm or norm-ish function on $\mathbb{R}^n$ (not ...

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**1**answer

3k views

### If then condition on mixed linear integer programming [closed]

Hi all. Let $a$ and $b$ be two real variables such that $0 \le a \le a_{max}$ and $0 \le b \le b_{max}$. I must write the following if-then-else condition with linear inequalities:
if $a < ...

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180 views

### A linear program related question

Dear all, recently, I encountered the following problem. It is closely related to the order of growth for UMD constants of all $n$-dimensional Banach lattice.
Let $\alpha^k \in (\alpha_1^k, ...

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71 views

### Computing maximum point for minimal function of a family of linear functions

Let $x \in S^n $ where $S^n = ${$ [x_1,x_2,...,x_{n+1}]\in \mathbb{R}^{n+1} \mid x \ge 0 , \sum x_i = 1 $} and let $f_i : I^n \to \mathbb{R}$ be a finite $m$-sized family of LINEAR functions such ...

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**1**answer

108 views

### Cascading minimization problems

Hi all. Suppose I have a linear programming problem on the vector variable $x$ that has many solutions and let $U$ be the set of these solutions. Suppose I have a second LP problem on $y \in U$. ...

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616 views

### linear programming with OR restrictions

Hi all. I have a linear program with the restriction that every variable can be zero or greater than or equal to a positive constant. That is:
minimize: $w^Tx$
subject to: $Ax=b$, $Cx \le d$ and for ...

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**2**answers

473 views

### Find both maximum and minimum values in linear programming problem

Hi all. I have a linear programming problem where I need to find both maximum and minimum values of the objective function. The optimal points are not relevant.
Is there an efficient way to do so?

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113 views

### LP/QP with not-so-constant linear constaints

I have an otherwise standard LP or PSD QP problem as below:
$\min\limits_x {c}' x$ subject to $Ax\leq b$
or
$\min\limits_x \frac{1}{2}{x}' Qx + {c}' x$ subject to $Ax\leq b$
the only exception ...

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204 views

### LP relaxation for ILP\IP (integer linear programming)

I am familiar with LP relaxation for ILP (or IP). Assume we concern with integer minimization problem, which we formalize using ILP; we then relax the ILP into LP and we say that the LP provides a ...

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**1**answer

281 views

### Solving a system of linear inequalities

Consider the following system of inequalities:
$Ax=b$;
$x\geq 0$;
A is a $m\times n$ (non-square) and sparse matrix in which some part of entries are rational. How this system can be solved without ...

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**1**answer

137 views

### SDP Algorithms/ maximally complementary solutions

Hello,
I was wondering if there are algorithms for (linear) Semidefinite Programs (SDP) out there, that converge towards a maximally complementary solution, even if strict complementary does not hold.
...

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**1**answer

246 views

### $\ell_o$ Minimization (Minimizing the support of a vector)

I have been looking into the problem
$\min: \|x \|_0$ subject to$: Ax=b$. $\|x \|_0$ is not a linear function and can't be solved as a linear (or integer) program in its current form. Most of my time ...

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votes

**1**answer

1k views

### sum of maxima vs the maximum of the sum

Consider the following integer program
$$
\begin{align}
\max &\sum\nolimits_{i}\sum\nolimits_{j} U_i(j)\cdot x_{i,j}\\
\text{subject to}& \sum_{i}x_{i,j}\cdot f\left(i,j\right)\leqslant ...

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vote

**2**answers

192 views

### what method can I employ to solve this optimization problem which involves \min?

The optimization problem is:
maximize $$\min(\sum\limits_{i=1}^N \log\left(a_{1,i}+\frac{b_{1,i}}{c_{1,i}+d_{1,i}x_i}\right),\sum\limits_{i=1}^N ...

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**2**answers

1k views

### Dual Norm For Sum of 2-Norms

What is the dual of a norm that is the sum of two-norms? Specifically, say we have the following norm for $\mathbf{x}\in \mathbb{R}^n$ and $\mathbf{A}_i \in \mathbb{R}^{m \times n}$
$\|\mathbf{x}\| = ...

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294 views

### constructing a curve dividing two sets of points

Lets assume I have two sets of points, each characterized as being "A" or "B", respectively, that are in a Euclidean plane. Theoretically these two sets are samplings from a space that has ...

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**3**answers

765 views

### Efficient Algorithm For Projection Onto A Convex Set

Given $\mathbf{x} \in \mathbb{R}^n$ and $\tau$ a scalar, I would like to solve the following Euclidean projection problem:
$\underset{\mathbf{p}}{\mathrm{argmin}} \; \|\mathbf{p}-\mathbf{x}\|_2
\;\;
...

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**0**answers

164 views

### Consistency of a system of linear equations

I have a system of linear equations in form of $AX=b$ where $A_{m\times n}$, $X_{n\times 1}$ and $b_{m\times 1}$. Coefficient matrix $A$ is quite sparse. However, using a practical LP solver like ...

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253 views

### Totally Uni-modular Matrices

A matrix is totally uni-modular if the determinant of any (square) sub-matrix is {+1, 0, -1}. My question is, "Is there a way to transform(linear or non) a general matrix into a totally uni-modular ...

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**1**answer

268 views

### Grading a non-graded poset as squeezed as possible

Here is a curiosity question (motivated by the recent revamp of ranked-poset routines in Sage).
Let $P$ be a finite poset. We look for a family $\left(a_p\right)_{p\in P}$ of real numbers summing up ...

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276 views

### Network flows with capacities on pairs of edges

Take a standard network flow problem: a directed graph with nonnegative capacities on each edge, a source $s$, a sink $t$. We all know how to find the maximum flow from $s$ to $t$.
Now add edge-pair ...

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**1**answer

130 views

### Continuity of Lexicographic Minimum Solution of a parametrized LP problem

Given a parametrized LP problem
find x, that minimizes F*x
such that Ax <=Bt+D
where t is a parameter.
And suppose C(t) is a set of all optimal solutions of LP with parameter t.
Let x_L(t) be ...

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**1**answer

414 views

### Algorithm for satisfiability of inequalities.

I am looking for an algorithm for checking the satisfiability (with natural values) of a set of inequalities made of variables and natural numbers, for example: $u < v, u \leq z, 3 \leq v$.
In ...

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**1**answer

178 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 ...

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**1**answer

195 views

### Is the Simplex Method still polynomial when all inequalities are through the origin?

Hello,
I want to solve a linear program using the simplex method, and I know that all my inequalities will pass through the origin (therefore, either my initial solution of (0, ... , 0) is optimal, ...

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**2**answers

494 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 ...

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**1**answer

229 views

### Checking if one polytope is contained in another

Hi,
I have two sets of inequalities, say, $Ax \leq 0$ and $Bx \leq 0$. I would like to know if they both define the same polytope. Or, even, whether one is contained in the other.
At the moment I am ...