All Questions
686 questions
1
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190
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linear relaxation of an optimization problem
I'm reading an article lately, and there is one point which confuses me.
So, we have the following constrained binary quadratic problem.
min $x^{T}Qx$
with the constraints that $Ax\leq b$ and $x\in ...
- 11
1
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0
answers
83
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Nonconvex Optimization of inner product objective
Does there exist any result on the following minimization,
$$\min_{x\in P} \langle x, F(x)\rangle\equiv \sum_i x_i F_i(x), $$ where $P$ is a convex polytope and $F_i(\cdot)$s are convex functions of $...
- 133
1
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0
answers
1k
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Number of different combinations in a 0-1 knapsack problem with integer weights [closed]
My question is actually very similar to this other one: Given a vector of positive integers, count the number of combinations which have a sum that produces a different value. But, since this previous ...
- 153
1
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0
answers
1k
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Analytic formula for minimizing the maximum inner product of a set of vectors
Given $x_j\in\mathbb{R}^n$, $j=1,\ldots,p$, find
$$
\widehat{w} \in \arg\min_{\Vert w\Vert=1}\max_{1\le j\le p} |\langle w,x_j\rangle|.
$$
I am also interested in the special case where we further ...
- 710
1
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0
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187
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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
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0
answers
83
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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 $B$...
1
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0
answers
232
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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
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0
answers
94
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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
1
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0
answers
64
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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
1
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0
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55
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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
1
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0
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90
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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
1
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0
answers
36
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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
1
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0
answers
88
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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
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1
answer
219
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approximate diameter of polytopes in high dimensions
I just came across the following problem:
Let us consider the unit corner of the n-cube
$$
\Delta^n = \left\{(t_1,\cdots,t_n)\in\mathbb{R}^n\mid\sum_{i = 1}^{n}{t_i} \leq 1 \mbox{ and } t_i \ge 0 \...
- 75
1
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0
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171
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Finding all feasible solutions
Let $u$ be a $n_{max} \times m$ matrix. Let $z$ be a $n_{max} \times s_{max} \times n_{max}$ cube. Let $w$ be a $n_{max} \times 1$ vector. All the three matrices can have values from the set $\{ 0, 1\}...
- 175
1
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0
answers
120
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The column generation technique on a Train Unit Assignment Problem [Linear Programming]
I am doing an assignment where I need to implement a mathematical model that I can't wrap my head around. For the technique of column generation, one would need to my understanding, a master problem ...
- 111
1
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0
answers
85
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Smallest sum of original column entries in 2d matrix [closed]
I have an interesting optimization problem I am trying to solve now and I thought I'd share it here in order to find the best answer. The problem itself is not complicated and it is stated like this:
...
- 111
1
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1
answer
6k
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Convert linear programming problem into its standard form [closed]
all,
I met a question that, the cost function of the linear programming problem is a function with absolute value. Here is the problem:
min 3x1+|6x2+3|
st.
|x1+4|+|2x2|<=3
How can I deal with it?...
- 13
1
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0
answers
100
views
Changing a nonlinear equality constraint into some conic inequality plus rank constraint
If we have a constraint optimization problem in which one of our constraint is $\prod\limits_{k = 1}^N {\left( {x - {a_k}} \right) = 0} $ . How could this nonlinear equality condition be changed into ...
- 33
1
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0
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140
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Reduce a Combinatorial problem
It is given n sets with k vectors. (k is element-wise positive or zero)
Choose one vector of each set so that the biggest element of the sum of the chosen vectors is minimal.
What i also know but is ...
- 11
1
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2
answers
172
views
Linear Programm with matrix [closed]
Is there a name for problems like this
min norm(Cx)
Ax = b
where C is a matrix and norm is the maximum norm.
This is kind of like a linear Programm. Could this be rewritten as linear programm? Or Any ...
- 11
1
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0
answers
75
views
Are there any known bounds on the value of solutions of linear integer programming?
Given a linear objective function and a system of linear constraints; are there any known bounds on the values of (positive) integral solutions in terms of the coefficient matrix of the constraints?
...
- 111
1
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0
answers
214
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range of the difference-of-two-qubit-$4 \times 4$-density-matrix-determinants
The determinant of a two-qubit $4 \times 4$ density matrix--that is, a Hermitian, nonnegative definite matrix with unit trace--lies between $0$ and $(\frac{1}{2})^8$. (A "pure state" has determinant ...
- 723
1
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0
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493
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Complexity of Nested Linear Optimization
My question is motivated by the fact, that among other ways, it is possible to restrict a variable to two discrete values, e.g. the prototypical $0$ and $1$, via an optimization constraint:
$$\max(\...
- 13.2k
1
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0
answers
196
views
Interior point optimisation using big M for L1 norm on linear system using Dikin's Affine method
I am a 4th year undergrad surveying student studying computations, specifically $L_{1}$ norm minimisation of residuals in large data sets. To start with (and probably to finish with) I'm using a set ...
- 111
1
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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
1
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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
1
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0
answers
1k
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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 ...
1
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0
answers
126
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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
1
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1
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460
views
Global maximization of a particular function
Hello!
I want to prove that $x = 0.5$ is the global maximum of the function
$f(x) = \frac{(1-a)^2e^{(2x\cdot(x-1)a^2)}}{(1-a)(e^{(2x\cdot(2x-1)a^2)}+e^{((2x-1)\cdot(2x-2)a^2)})-2(1-2a)e^{(4x\cdot(x-...
- 43
1
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0
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628
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Totally unimodular 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 ...
- 11
1
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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 ...
- 2,966
1
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0
answers
1k
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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 u_{...
- 53
1
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0
answers
268
views
On the convergence of a special fixed point iteration
The problem is actually a quadratically constrained quadratic program. And the formulation is:
$max: \frac{1}{2}x^TQx + d^Tx$
$s.t. x\in R^{n,+} ,\sum_{i\in I_p}x_i^2=1, p=1..k$
where $d\in R^{n,+}$,...
- 11
1
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0
answers
1k
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Covariance matrix formula interpretation - what am I missing?
I'm reading a paper that outlines the calculation of a covariance matrix like the following:
$C=\displaystyle\sum^{N_b}_{i=1}\vec{x}_i\vec{x}_i^T$
What is the order of this matrix? My interpretation ...
- 111
0
votes
1
answer
1k
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For Ax = b, x and b unknown vectors, how do I solve the x that maximizes min(b_i)?
Given a matrix $A$, each element $A_{i,j} \geq 0$, find the vector $\vec x$ that maximizes the minimum element in $\vec b$ ($\vec b = A \vec x$). Note that this is not a linear equation system as I ...
- 135
0
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1
answer
319
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Correct way to conduct equilibrium scaling of linear/integer/MIP program
I would like to scale my linear/integer program and also mixed-integer program using the equilibrium scaling method. I have worked on two research papers and one research book. However, they did the ...
- 21
0
votes
1
answer
539
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: $$f(p_{1,i},p_{2,i},\dotsc,p_{m,i})=\sum_{i=1}^{n}\begin{bmatrix}p_{1,i} & p_{2,i} & \...
- 3
0
votes
1
answer
226
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Fractional values in linear programming
Consider the linear programming problem
\begin{align}
f^* = \max_{x}&~p^Tx~\\~&A^Tx\leq b~,~0\leq x_i\leq 1
\end{align}where $p$ is a $n\times 1 $ vector, $A$ is a $n\times c$ matrix and $b$ ...
- 1,421
0
votes
1
answer
144
views
Maximize function on rotation matrices [closed]
Let $A$ be a fixed 3-by-3 matrix and $Q$ be a rotation matrix whose yaw, pitch, and roll angles are $\phi\in[0,\pi]$, $\theta\in[0,\pi]$, and $\psi\in[0,\pi/2]$, respectively:
\begin{equation}
Q=
\...
0
votes
1
answer
150
views
Differences between the convex discrete maximization and minimization problems? [closed]
Would you tell me some main distinctions between the convex discrete minimization and maximization optimization problems?
In the case of the feasible are bounded then we only need to transform one ...
- 17
0
votes
1
answer
36
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Benefit of adding a trivial constraint to ILPs
let ILP be an integer linear program with constraints-matrix $\boldsymbol{\mathrm{M}}\in\mathbb{Z}^{m\times n}$ and cost vector $\boldsymbol{\mathrm{c}}\in\mathbb{Z}^n$,
${\boldsymbol{\mathrm{x}}^*}\...
- 13.2k
0
votes
1
answer
93
views
How quickly can this IQP or its MILP relaxation be solved
Let $A\in\{0,1\}^{(n,n)}$ be a $n$ by $n$ boolean matrix (in particular think of an adjacency matrix of a graph), and consider the following optimization problem:
$$\begin{align*}&&\max_{P\in\{...
- 71
0
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1
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131
views
How hard is a linear programming with a bounded constraint?
Background: I am reading Greg Kuperberg's answer to the question Deciding membership in a convex hull. I am thinking about the complexity of ''Deciding membership in a convex hull''.
Restate the ...
0
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1
answer
214
views
How do you call a linear programming problem when the solution should be "constrained" to a norm?
(apologies for the n00b question)
Let's say we have a vector of length $n$, with to-be-determined values: $a_1, a_2, ...,a_n$.
And we have information that partial sums of these elements are equal to ...
- 143
0
votes
1
answer
320
views
Sub optimal algorithm for linear programming
Consider the linear programming problem
\begin{align}
f^* = \max_{x}&~p^Tx~\\~&A^Tx\leq b~,~0\leq x_i\leq 1
\end{align}where $c$ is a $n\times 1 $ vector, $A$ is a $n\times c$ matrix and $b$ ...
- 1,421
0
votes
1
answer
39
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Gluing simplices through a common point/ realisation of a convex simplicial polytope
Given $m≥d+1$
a positive integer, is it always possible to find m d-dimensional simplices $\Delta_i=\mathrm{Conv}(M,V_{i,1},…,V_{i,d})$ such that
1) they all share the common vertex M
2) the ...
- 1,435
0
votes
1
answer
212
views
How to find out if a polytope contains a sphere?
Given a polytope described by linear inequalities $Ax \le b, x \in \mathbb R^n$, how do you find out if there exist a (non degenerate) sphere of dimension $n-1$ contained in the polytope?
Thanks!
- 225
0
votes
1
answer
130
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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 $y \in U$. ...
- 57
0
votes
2
answers
891
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?
- 57