Questions tagged [linear-programming]

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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Implementation of Friedman's algorithm of reconstructing simple polytopes

In Finding a Simple Polytope from Its Graph in Polynomial Time, Friedman gave a polynomial time algorithm on reconstructing a simple polytope from its graph. Has this algorithm been actually ...
mashedcarrots's user avatar
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Max-flow modeling with unified vehicle and commodity variables

I am working on a network flow problem that involves routing through a time-space network. The network consists of: A single source node and a single demand node. A fleet of vehicles with specified ...
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Application of greedy approach for optimization

I want to maximize an objective given by $$\max_{\{q_n,p_n\}} \sum_{n=0}^\infty (\alpha_1 - \beta_1 n) p_n + (\alpha_2 - \beta_2 n) q_n$$ where $\alpha_1 > \beta_1 >0$ and $\alpha_2 > \beta_2 ...
Prakirt Raj's user avatar
2 votes
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Is matrix B obtained from matrix A?

Assuming a matrix $\mathbf{A} \in \mathbb{R}^{4096 \times 4096}$ sampled from a standard normal distribution $N(0, 1)$, and another matrix $\mathbf{B} \in \mathbb{R}^{4096 \times 4096}$ either sampled ...
eternity's user avatar
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ILPs with square constraint matrix

Given the Integer Linear Programming ($\text{ILP}$) problem \begin{array}{ll} \text{minimize} & c^T x \\ \text{subject to}& \mathbf{A}^T x \ge b \\ \text{where}&c,x,b\in\mathbb{N}_0^n,\\ &...
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Monotony of enforced subtour merging

Is it true that for a symmetric TSP instance in the sequence of edges generated by successively: calculating the optimal 2-factor adding cardinality constraints on the edgesets of the 2-factor's ...
Manfred Weis's user avatar
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Solve NP-hard type problems with linear programming

I would like to know if there is any way to solve an NP-hard type problem, for example, the TSP, sum of subsets or knapsack problem, by using linear programming and not by brute force. I ask this ...
Juan Carlos's user avatar
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Alternatives to McCormick Envelope

I have an optimization problem for which I have the optimal solution obtained by the ILP. However, when I introduced the McCormick Envelope to replace the product of a bi-linear term in its LP ...
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Seeking insights on bounded set positive solutions for a set of linear systems in $\mathbb{R}^n$

Before delving into my query, I'd like to provide some context. Consider a continuous function $f:\mathbb{R}^{k}\rightarrow\mathbb{R}^{m}$ and a compact set $\mathcal{B}\subset \mathbb{R}^{k}$ (...
Diego Fonseca's user avatar
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How to integrate an indicator function/constraint into the cost function of a linear program?

I have a mathematical model $P$ for which I optimize two cost functions say $F_1$ and $F_2$ subject to a set of constraints $C1$–$C10$. In $F_2$, I want it to be included only when its expression ...
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Inf-convolution of norm 1 and norm 2 square

The inf-convolution of the functions $f$ and $g$ defined on $\mathbb{R}^n$ is $$ h(x)=\inf _{y \in \mathbb{R}^n} f(y)+g(x-y) . $$ We can prove that if $f,g$ are convex functions, then $h$ is convex. ...
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Calculating vertex potentials from optimal matchings

Question: can the solution to the dual of a Linear Program be calculated directly from the solution of the primal Linear Program? If yes, what are known algorithms and their bounds on complexity. As ...
Manfred Weis's user avatar
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An $n$-dimensional generalized Hoffman’s circulation theorem?

For a directed graph $G$, a 1-dimensional circulation is a function $f:E(G)\rightarrow \mathbb{R}$ such that for every $v\in V(G)$, $$\sum_{uv\in E(G)}f(uv)=\sum_{vw\in E(G)}f(vw),$$ where $uv$ is an ...
Connor's user avatar
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Is the problem of vertex enumeration from an H-representation of a polytope NP-hard?

According to the Wikipedia page on the issue, the vertex enumeration problem is NP-hard. However, double description and reverse linear search are algorithms listed to solve the problem. Moreover, ...
Makogan's user avatar
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Relationship of optimal solutions between the total function and the sub function

This is an unconstrained convex optimization problem. Let $\mathcal{N}=\left\{1,\ldots,n\right\}$, $2\leq n<\infty$. Suppose there are many strongly convex functions $f_i(x)$, where $x\in\mathbb{R}^...
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Show that $\max_{P_X : X\in (0,1) } \left| \frac{\mathbb{E} [ f'(X) ]}{ \mathbb{E} [ f(X) ] } \right|$ is maximized by at most two mass points

Let $f$ be some given well-behaved function. Consider the following optimization problem overall probability distribution on $[0,1]$ \begin{align} \max_{P_X : X\in [0,1] } \left| \frac{\mathbb{E} [ ...
Boby's user avatar
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$A^*$ algorithm to find shortest path when weights in my graph are the inverse of distance

Given a graph G=(V,E) where the weights on my edges are inverse of Euclidian distance between nodes, I want to know if I can use A* algorithm to find the shortest path. How I need to modify the ...
user2512443's user avatar
1 vote
1 answer
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Optimization on non-convex set

Let $\Omega$ be an open bounded subset of $\mathbb{R}^2$ and $f\in L^2(\Omega)$ be a given function. Consider the optimization problem $$\mathrm{min} \int_\Omega u(x) f(x) \,dx\,,$$ where a minimum is ...
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Basis of monoid of integral vectors

Suppose that $M\in\mathbb{Z}^{n\times k}$ is a matrix of rank $k<n$. How can I obtain a set of vectors $b_1,\ldots,b_k\in\mathbb{Z}^k$ (if exists) such that each row of $M$ is a non-negative ...
Brauer Suzuki's user avatar
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LP formulation of $k$-opt moves

Question: what is known about formulating $k$-opt moves that strive for improving the length of Hamilton cycles by means of exchanging $k$ of the tour edges with $k$ non-tour edges? Specifically: are ...
Manfred Weis's user avatar
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Any idea of solving an optimization problem with cubic constraints?

I have the following optimization problem with cubic constraints, which is hard to solve. Are there any ideas, or related references, of solving such a problem? $$ \begin{array}{ll} \underset {y, z} {\...
Erik's user avatar
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How to solve the optimization problem $\max_{\mathbf{w}}\sum_i\text{sign}(\mathbf{w}^T \mathbf{x}_i)$?

I am looking for an algorithm to solve the following optimization problem $$\max_{\mathbf{w}}\sum_i\text{sign}(\mathbf{w}^T \mathbf{x}_i)$$ where $\mathbf{w}$ and each $\mathbf{x}_i\in\mathbb{R}^d$. ...
user3750444's user avatar
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Techniques for solving linear inequalities

For $n$ real variables $x_1, \ldots, x_n$, I have a bunch of inequalities of form $2 x_i > x_j + x_k$ or $2 x_i < x_j + x_k$, where $i,j,k$ are distinct. My goal is to determine whether this set ...
Dmitry's user avatar
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Constrained linear optimization problem on $C^1$

I am dealing with a problem of the form ($a<b$) $$ \displaystyle \max_{v \in C^1([a, b])} \int_a^b v(x)~\mathrm{d}x, \quad \mathrm{s.t.} \int^b_a \big(-o'(x)v(x)-v'(x)o(x)\big)f(x)~\mathrm{d}x \...
Hyperbolic PDE friend's user avatar
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Does Hoffman constant keep the same after a very tiny perturbation on the polyhedron such that the bases are even unchanegd?

Suppose that $P$ is a polyhedron represented by $$P:=\{x \in \mathbb{R}^n: A x \le b \} \text{ for }A \in \mathbb{R}^{m\times n},\ b \in \mathbb{R}^m,$$ and $P$ contains interior points. Moreover, the ...
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Bound on solutions of $Ax \ge b$

Let $A \in \mathbb{Z}^{m \times n}, b \in \mathbb{Z}^{m \times 1}$. One can show that if there is a solution of $Ax \ge b, x \in \mathbb{R}^n$ then there is one such that $\|x\|_{\infty} \le c (\|A\|_{...
user1868607's user avatar
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1-degree SOS proof refutes Linear Programming

I am trying to understand Sums-of-Squares proof systems. A degree $d$ Sums-of-Squares refutation for a set of polynomial equations $P = \{p_1(x) = 0, ..., p_m(x) = 0\}$ is defined as $\sum_{i=1}^m g_i(...
Tom Keaton's user avatar
1 vote
1 answer
97 views

Adding linear constraint to the domain

I don't know if it is a well-known problem, but I have been struggling to come up with an algorithm. I have a set of linear constraints $Ax\le b$, $b\ge 0$ ($b$ and $A$ are given, $x$ is a variable). ...
Ryszard Eggink's user avatar
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On optimizing a multivariate quadratic function subject to certain conditions

The problem is to maximize $f(x_1,x_2,\cdots,x_n)=\sum\limits_{i=1}^{n}\Big(x_i-k_i\Big)^2$ for $n\ge 3$ subject to the conditions (1) $\sum\limits_{i=1}^{n}x_i=\sum\limits_{i=1}^{n}k_i\le n(n-1)$ ...
shahulhameed's user avatar
1 vote
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232 views

Closed-form solution of a particular linear program

(Note: I asked a similar question at math.stackexchange but the present one is more precise.) I have a linear program of the form: $$\text{minimize} \space\space x_1 \space\space \text{subject to:}$$ $...
Fabius Wiesner's user avatar
1 vote
1 answer
118 views

Best projection on non-convex discrete set with two constraints

I want to compute the projection of a vector $\left( x\right) _{1\leq i,j\leq n}\in \lbrack 0,1]^{n\times n}$ on the following discrete set $$ S=\left\{ x\in \{0,1\}^{n\times n}:x_{i,j}+x_{j,i}\leq 1;\...
Goga's user avatar
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2 votes
1 answer
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Counting the number of pair of d-uplets with upper bounded distance

Consider two d-uplets $u = (u_1,...,u_d)$ and $v = (v_1, ..., v_d)$ both living in $\mathbb{N}^d$ with $d$ a positive integer. They both verify $$(*) \sum_{i=1}^d u_i = \sum_{i=1}^d v_i = k$$ with $k$ ...
Ludwich's user avatar
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1 answer
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Is there a redundant constraint in linear programming? [closed]

From wikipedia: But... Why do we need the $x\ge 0$ part? We can instead do $-x\le 0$, and thus saving a line in the definition (which is not a big deal but nevertheless nice). (In order to do that, ...
Bipolo's user avatar
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0 answers
234 views

Finding the eigenvectors of a submatrix

Let $A=(a_{kl})$ be a matrix in $M_n(\mathbb{R})$ when $n$ is even. Let $B=(b_{kl})$ be the symmetric $2n$ by $2n$ matrix whose entries are given by, $b_{k,l}=a_{kl}$ if $1\leq k,l\leq n$. $b_{n+k,l}=...
ABB's user avatar
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Nash Equilibria change linearly in (some) game parameters. Already known / follows from a more general result?

EDIT: The key thing that I am wondering about is the linearity of the P2 strategy, not the constancy of P1. (The latter is straightforward.) Question: Is the following result already known? Or is it a ...
Vojtěch Kovařík's user avatar
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1 answer
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Combining Dantzig-Wolfe and Benders decomposition

I'm now solving an LP that has a few coupling rows (as in Dantzig-Wolfe decomposition) and a few coupling columns (as in Benders decomposition) simultaneously; other rows and columns are block-angular....
tlu's user avatar
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1 answer
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Benefit of adding a trivial constraint to ILPs

let ILP be an integer linear program with constraints-matrix $\boldsymbol{\mathrm{M}}\in\mathbb{Z}^{m\times n}$ and cost vector $\boldsymbol{\mathrm{c}}\in\mathbb{Z}^n$, ${\boldsymbol{\mathrm{x}}^*}\...
Manfred Weis's user avatar
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Boolean operation on n dimensional polyhedron

A polyhedron in $R^n$ is defined by a set of half-planes: $P = \{x \in R^n \mid Ax - b \le 0\}$. Given a set of polyhedra in $R^n$, $ P_1, P_2, \dotsc, P_k$, is there an algorithm/implementation that ...
Robin Lee's user avatar
0 votes
1 answer
78 views

How quickly can this IQP or its MILP relaxation be solved

Let $A\in\{0,1\}^{(n,n)}$ be a $n$ by $n$ boolean matrix (in particular think of an adjacency matrix of a graph), and consider the following optimization problem: $$\begin{align*}&&\max_{P\in\{...
alosc's user avatar
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1 answer
260 views

Method for (binary) optimization under constraints

I would like to know if there is a method to solve the Problem. Problem: Maximize the following function: $$f(p_{1,i},p_{2,i},\dotsc,p_{m,i})=\sum_{i=1}^{n}\begin{bmatrix}p_{1,i} & p_{2,i} & \...
kris's user avatar
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1 vote
0 answers
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How do I incorporate Ito's lemma into the solution for a finite-horizon stochastic cake-eating problem?

I'm interested in finite-horizon, continuous-time cake-eating problems in which the agent has a time-horizon $W$ over which to eat the cake, and then chooses an optimal consumption path $\{h_t\}_0^W$, ...
C_A_Pepe's user avatar
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1 answer
99 views

$\mathrm{ILP}$-formulation for Minimum Maximal Matching (MMM) Problem

Despite some online searching I couldn't find examples of dedicated Integer Linear Programs ($\mathrm{ILP}$s) for determining smallest matchings, that are not contained in a larger one. It seems that ...
Manfred Weis's user avatar
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1 vote
1 answer
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Linear programming with "nice" matrices

Consider the following linear programming problem \begin{array}{ll} \text{minimize} & \mathrm 1^{\top} \mathrm x\\ \text{subject to} & v\le \mathrm A \mathrm x \le u\\ & \mathrm x \geq ...
user12345678's user avatar
1 vote
1 answer
549 views

Interpreting mincost flow dual variables

Consider the task of finding flow of size $b$ with minimum possible cost. It may be formulated as linear programming in a following way: $$\boxed{\begin{gather} \min\limits_{f_{ij} \in \mathbb R} &...
Oleksandr  Kulkov's user avatar
0 votes
1 answer
62 views

Round Robin volleyball Tournament [closed]

Consider a set of N teams (N even number) that must make a Round Robin Tournament. To each pair i; j, i ≠ j, of teams there is associated level of interest si,j ∈ {1;2;3} of the match between them (1 =...
Giuseppe Teodoro's user avatar
2 votes
1 answer
222 views

Solving linear programming without solving linear programming

Let $v_1, \cdots, v_n$ be vectors in $\mathbb R^k$, and let $M$ be the Gram matrix of them. It's possible to determine from $M$ and $k$ whether the only vector that has nonnegative inner product with ...
LeechLattice's user avatar
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2 votes
1 answer
333 views

Who called Farkas' fundamental theorem a lemma?

Farkas proved his famous result (which, nowadays, is fundamental in optimization theory) in 1902 and called it Grundsatz der einfachen Ungleichung which may be translated as fundamental theorem of ...
Jochen Wengenroth's user avatar
1 vote
1 answer
284 views

Finding a special solution in a solution set over F2

Given a solution set of a linear system of the following form $$ \{ \begin{bmatrix} x_{1} \\ \vdots \\ x_{n} \end{bmatrix} = \vec{v_1} * x_1 + \dots + \vec{...
borekking's user avatar
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1 answer
245 views

What is the best way to choose initial basis when applying simplex method to an equality form of LP?

Currently I'm trying to write a practically fast LP solver for a sparse instance, which is by simplex method with LU decomposition and eta-matrix update. In the development I realized that I'm not ...
sansaqua's user avatar
4 votes
2 answers
274 views

Connecting $2n$ points in $\mathbb R^2$ with line segments s.t. each point belongs to exactly one line segment

I'm trying to do a certain simulation related to the toric code and I'm looking for an algorithm that connects $2n$ points ($n \in \mathbb Z_+$) in $\mathbb R^2$ with line segments with the following ...
Sanchayan Dutta's user avatar

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