Linear programming is the study of optimizing a linear function over a set of linear inequalities. The Simplex Method, Ellipsoid Method and Interior Point Method are popular algorithms to solve linear programs.

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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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291 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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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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1answer
360 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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2k 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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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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2answers
2k 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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2answers
357 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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3answers
1k 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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172 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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274 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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275 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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292 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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1answer
150 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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1answer
469 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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190 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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1answer
214 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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2answers
630 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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1answer
260 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 ...
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1answer
109 views

Design constraint systems over the reals

This question is inspired by the discussion at this problem. Suppose I have a design consisting of a finite point set $U$ of size $|U|=m_{\emptyset}$ and a family of $n$ subsets (sometimes called ...
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2answers
925 views

how to model a linear program with step-like cost function in the objective

I have a large linear program with the following details. d1 to di are the variables, where di is an integer. The constraints are a series of inequalities of the form d1 < d3 < d7 < d23 ...
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How do you tell if a system of linear inequalities has a solution?

A naive solution would be to optimize a dummy variable via linear programming and see if a result is returned. I imagine there must be a more direct way.
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0,1 solution to system of linear integer equations.

I have the following problem: $A x = b$ where $A, b$ - $m \times n$-maxtrix and $m$-vector of nonnegative intgers (respectivelly). $x \in \{0,1\}^n $ - vector of binary variables, which need to be ...
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Could SVD be used to optimize the partial inner-products?

Suppose a set $N$ of $n$ distinct points in $m-$dimensional space is given in $X\in\mathbb{R}^{n\times m}$. Also, suppose a subset $L\subset N$, $|L|=l<m<n$, with $m-$dimensional coordinates in ...
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Does the Hirsch conjecture hold for $n < 2d$?

The Hirsch conjecture asserts that the graph (i.e. $1$-skeleton) of a $d$-dimensional convex polytope with $n$ facets has diameter at most $n - d$. After being open for decades, Francisco Santos has ...
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2answers
1k views

Solving a system of equations/inequalities that have trigonometric functions on the left-hand side

Is there any known (symbolic) method that solves a system of equations/inequalities that have trigonometric functions on the left-hand side of the system? Ex) Find $x,y,\theta \in \mathbb{R}$ that ...
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Closest 3D rotation matrix in the Frobenius norm sense

Given a 3 by 3 matrix $M$ I would like to find the rotation matrix $R$ minimizing the Frobenius norm: \begin{equation} \|R-M\|_F \end{equation} Is there a closed form solution for $R$, or is it ...
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1answer
190 views

Conditions for differentiability of minima and minimizers of linear functionals?

Let $B$ be a (real) Banach space and $C$ be a compact convex subset of $B$. For every continuous linear functional $F$ on $B$, define $V(F)=min_{c\epsilon C} F(c)$ and $S(F)= { \lbrace c \epsilon C ...
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483 views

Solving for Hamiltonian path with constraints on allowable routes through vertices

Suppose you have a complete graph with N vertexes, with a distinguished vertex $n=1$ ("start"), and you wish to find a route traveling exactly once through each vertex so that the distance along the ...
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1answer
391 views

Minimum distance between two data sets

Suppose we have two sets of data, $X$ and $Y$, each of which contains $10$ positive numbers. Now let us order the data sets $X=\left\{ x_{1},\cdots,x_{10}\right\}$, $x_{1}\ge\cdots\ge x_{10}>0$ and ...
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How to solve simple bilinear equations under extra linear constraints

Hello, This is the full version of a question I asked earlier. I am trying to understand whether finding a solution to the following bilinear system is computationally hard or easy: $\lambda_i^T ...
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1answer
727 views

Can one efficiently optimize over the inverse of matrix?

Hello, I have the following problem: Find a non-negative matrix $L$ (i.e. $L_{i,j} \geq 0$ for all $i,j$), $L \neq I$ so that $A(I-L)^{-1}y \geq 0$ (the inequality must hold for each component), ...
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114 views

sparsest cut always has solution

Hi! How to prove that sparsest cut always has an optimal solution which is the cut for some vertex-subset. Looks like it's should be a kind of fundamental theorem for sparsest cut. But I didn't ...
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1answer
243 views

Approximate Set Cover Problem by Rounding

Here is the simple algorithm for approximating set cover problem using rounding: Algorithm 14.1 (Set cover via LP-rounding) Find an optimal solution to the LP-relaxation. Pick all sets ...
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1answer
109 views

Are there intuitive/classically algorithmic analogues to Semidefinite programs on networks?

Many network optimization algorithms, including shortest path, push-relabel, augmenting path, etc, actually have an interpretation in terms of linear programming. A famous application of semidefinite ...
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1answer
385 views

Need help to find an efficient algorithm for the following problem!

Consider $x$ an n-dimensional vector with $x_i$ is integer in the range $[0 \dots k], k\in N$. Given $A_{n\times n}$ is the covariance matrix of $x$. $u$ is a given n-dimensional vector of real ...
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256 views

Lovasz theta function - uses

Lovasz theta function bounds the Shannon capacity of graphs. What are some other uses of the function - especially in asymptotic coding theory and optimization problems?
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1answer
293 views

Partially optimal solutions in integer linear programming

Linear programs with a totally unimodular system matrix are known to have an optimal integer point. They are therefore solvable via relaxing the integer constraints to intervals. An other interesting ...
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619 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 ...
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539 views

Guessing game with guess cost

This is a question about Problem 328 on the website Project Euler. A description of the problem is provided in the previous link. I was wondering if there has been any research done on this question. ...
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1answer
545 views

Complexity of a weirdo two-dimensional sorting problem

Please forgive me if this is easy for some reason. Suppose given $S$, a set of $n^2$ points in $\mathbb{R}^2$. I want to choose a bijective map $f$ from $S$ to the set of lattice points in $\lbrace ...
2
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0answers
246 views

Linear complementarity problem: principal pivoting algorithm

I'm trying to implement the "Dantzig; van de Panne and Whinston" principal pivoting algorithm for solving symmetric positive semi-definite LCPs from "The Linear Complementarity Problem" book ...
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1answer
319 views

Mathematical Programming with other Algebras than Linear

Linear Programming is strongly entwined with linear algebra, as are many of its generalizations under the heading of mathematical programming / convex optimization. What analogies are there for ...
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Linear programming - uniqueness of optimal solution

Is it possible to build such an objective function for a given set of constraints, so that there will be only one optimal solution? My general problem is to get any vertex of a polytope formed by a ...
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Random Sampling a linearly constrained region in n-dimensions…

Hi, So here is my problem: Given a nonlinear, discontinous, cost function $f(x_1,x_2,..,x_N)$ along with linear constraints $x_n \ge 0, \forall n$ $x_n \le c_n$ and $\sum_{n=1}^N x_n = 1$ find an ...
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Number of breakpoints in parametric maximum flow problems

The parametric maximum flow problem can be formulated as $$f(\lambda) = \min_{x\in\{0,1\}^n} \left( \sum_{i}(a_i + b_i\lambda)x_i + \sum_{i,j}c_{ij}x_ix_j \right), $$ where all $c_{ij}<0$ (so that ...
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2answers
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Linear program to maximize the minimum absolute value of linear functions ?

I'd like to compute $\max_{x,t} t$ such that $\forall i$, $t < a_i + |x - b_i|$. where $a_i,\ldots, a_n$ and $b_1,\ldots,b_n$ are fixed and $x \in [0,1]$. Can this be solved with a linear ...
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6answers
1k 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 ...
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184 views

A polytope associated with the Hadamard Transform

In an investigation of whether or not a subset $V$ of "Hamming Space" $M_n = \mathbb{F}_2^n$ is a tile (i.e. whether $M_n$ can be written as a disjoint partition of translates of $V$) in ...
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641 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 ...