All Questions
Tagged with linear-programming oc.optimization-and-control
92 questions
1
vote
0
answers
81
views
Maximizing sum of homogeneous functions of order one over a polytope
Let $f_i: \mathbb{R}^n\rightarrow \mathbb{R}$ be
concave, increasing (i.e., if $x\geq y$ where the inequality is entry wise, we have $f_i(x)\geq f_i(y)$), and a
homogeneous function of order one for ...
- 393
2
votes
0
answers
105
views
Optimization over a convex cone generated by a set is equal to optimization over the set
Within my research I found an important doubt and that prevents me from advancing, the context of my doubt is as follows:
We considerer the following optimization problem
$$
\left\{\begin{array}{cl} \...
- 159
3
votes
1
answer
368
views
Lot sizing problem: how to add these cuts efficiently
Consider the set of constraints of the uncapacitated lot sizing problem:
$$
\{(x,s,y)\in \mathbb{R}^n_+ \times \mathbb{R}^n_+ \times \mathbb{B}^n \;|\;s_{t-1}+x_t = d_t+s_t,\; x_t \le My_t,\; t=1,\...
- 225
0
votes
2
answers
120
views
Reference request: dependence on linear constraints
Excuse me if my question is stupid. I'm seeking the references on the dependence of the (linear) optimization problem on (linear) constraints. Namely, consdier the following optimization problem:
$$P(...
- 1,835
3
votes
0
answers
71
views
Dependence of optimization problem on the linear constraints
Let $I=\{x_1,\cdots, x_n\}\subset \mathbb R$ be fixed. Given two probability distributions $\alpha=(\alpha_i)_{1\le i\le n}$ and $\beta=(\beta_i)_{1\le i\le n}$ on $I$, and a matrix $c=(c_{i,j})_{1\le ...
- 1,835
1
vote
0
answers
1k
views
Analytic formula for minimizing the maximum inner product of a set of vectors
Given $x_j\in\mathbb{R}^n$, $j=1,\ldots,p$, find
$$
\widehat{w} \in \arg\min_{\Vert w\Vert=1}\max_{1\le j\le p} |\langle w,x_j\rangle|.
$$
I am also interested in the special case where we further ...
- 710
1
vote
0
answers
187
views
Strong Duality of Mixed Integer Linear Program
The problem at hand is to optimize a mixed-integer linear program closely related to the maximum flow problem. I would like to reformulate the problem with its dual and I'm concerned with the ...
- 11
2
votes
0
answers
64
views
Finding orthogonal basis with constraint
Is there any fast algorithm that output an orthogonal basis $e_i,i\leq n$ of $R^n$
with $e_i\in V_i$? Where $V_i,i\leq n$ are given linear subspaces of $R^n$.
And is there any condition on $V_i,i\leq ...
- 909
0
votes
1
answer
204
views
Is the linear production game a convex game?
In cooperative game theory, the linear production game (LPG) is defined by letting the characteristic function have the form of a linear programming problem.
Does anyone know if the LPG is a convex ...
- 17
4
votes
1
answer
891
views
Basic result in semi-infinite linear programming
Consider a standard linear program of the form $$\textrm{minimize}_x~~~~ c^Tx~~~~ s.t. \\ Ax = b \\ x \geq 0$$ with $x\in \mathbb{R}^n$ and $A \in \mathbb{R}^{m \times n}$. It is well known that, if ...
- 4,049
6
votes
0
answers
97
views
Finding the optimal mixture of two convex functions
I am trying to find an efficient way to solve the problem $$\min_{p,x_1,x_2} p\cdot f(x_1)+ (1-p) \cdot f(x_2)~~~~~ s.t.\\p\cdot g_1(x_1) + (1-p)\cdot g_2(x_2)\leq 1 \\ 0\leq p \leq 1$$ where $x_1,x_2\...
2
votes
1
answer
104
views
Standard names and methods for this type of fitting minimization
In material science research, we have come across the following type of problem.
Given a m by n matrix A, a m vector b, and error tolerance $\varepsilon$, we want to do this minimization
$$\eqalign{
...
- 867
1
vote
1
answer
1k
views
Linear programming with exponentially many constraints and variables [closed]
From class, I learnt that problems like traveller salesman have a Linear programming representation with exponentially many constraints. Using method of separation, this problem is solved rather ...
- 867
0
votes
1
answer
270
views
Generalized assignment problem with no integrality gap
Suppose I am solving the generalized assignment problem, so that I
am given matrices $U$ and $W$ and a vector $c$ (all three of which
have, say, positive entries), and I want to solve
$$\text{...
0
votes
2
answers
708
views
Approximate solution to large mixed integer programming problem
What are the available approaches to find an approximate solution to a large mixed integer programming problem?
I ran my problem in the Gurobi MIP solver.
It can find a feasible solution in ...
user24451
2
votes
1
answer
191
views
Optimization problem whose cardinality never exceeds 7 for some reason
I am working on a problem in which I have a collection of $n$ points, $x_1,\dots,x_n$, in the plane, as well as a positive definite matrix $\Sigma$ and another point $\mu$ in the plane. I am trying ...
1
vote
1
answer
4k
views
Maximizing linear objective function with absolute values
This has be asked on other forums, though couldn't
find authoritative answer.
I have a linear program over the reals and don't
want to introduce integer or binary variables.
The objective function ...
- 25.4k
2
votes
2
answers
840
views
Finding the maximum of a multivariate polynomial of degree one
I need to find the global maximum of the function
\begin{align}
f\left(x\right) & = p_1 \max\left(\sum a_{1i} x_{1i}, \sum b_{1i} x_{1i}\right) - \sum c_{1i} x_{1i} \\
&+\ldots \\
&+ p_n ...
user24451
1
vote
0
answers
493
views
Complexity of Nested Linear Optimization
My question is motivated by the fact, that among other ways, it is possible to restrict a variable to two discrete values, e.g. the prototypical $0$ and $1$, via an optimization constraint:
$$\max(\...
- 13.2k
4
votes
3
answers
1k
views
Minimax theorem on a non convex domain
A minimax theorem is a theorem which states that under certain conditions on $\mathcal{X}$, $\mathcal{Y}$ and $f$:
$$ \inf_{x \in \mathcal{X}}{\sup_{y \in \mathcal{Y}}{f(x,y)}} = \sup_{y \in \mathcal{...
- 591
0
votes
1
answer
99
views
generalization from linear programming solution [closed]
I have a series of similar linear programs that depend on an input vector $a\in A$ and whose solution is an output vector $b\in B$. I can solve them individually, but this is wasteful. I suspect that ...
- 109
4
votes
2
answers
2k
views
Simplified knapsack problem
There is a problem that I can not solve.
Given a set of items (each item has some integer weight) we have to fill bag with some number of copies of these items, with the only restriction that the ...
- 41
0
votes
1
answer
85
views
About the suboptimality of linear estimators
Let $X$ be a random variable and $N$ a Gaussian noise independent from $X$. We observe $Y=X+N$ and want to estimate $X$ based on $Y$ to minimize the mean square error $mmse(X|Y):=E(\hat X(Y)-X)^2$.
...
- 29
0
votes
1
answer
2k
views
eigen-decomposition solution? is it unique?
Assume an N*N covariance matrix (Q) which is a positive definite matrix. The decoder X is assumed to be N*s, where s<=N. X is calculated to be s eigenvectors corresponding to s minimum eigenvalues. ...
- 1
5
votes
2
answers
888
views
relation between solution of a linear program and its perturbation
I have a linear program over a finite set of points $(x_1, x_2,\ldots, x_m)\in\mathbb{R}^n$:
$$
\max_j c' x_j
$$
Suppose the solution of this LP is obtained at a point $x_{j_1}$, which is a vertex ...
- 373
1
vote
2
answers
134
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 ...
- 11
10
votes
1
answer
411
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 ...
- 37.7k
1
vote
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
3
votes
2
answers
2k
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 ...
- 113
4
votes
1
answer
966
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 ...
2
votes
1
answer
130
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 ...
- 2,383
5
votes
3
answers
1k
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 ...
13
votes
2
answers
664
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 ...
- 19.2k
3
votes
1
answer
357
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 ...
- 2,383
3
votes
2
answers
5k
views
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 ...
- 500
4
votes
1
answer
2k
views
solving multiple linear programming problems with the same set of constraints
Hi,
I need to solve a set of linear programs of the form:
Problem $i$: $\quad \max c_i \cdot x$ s.t. $ A x \leq b$.
The $c_i$'s are different vectors so each problem has a different objective ...
- 560
1
vote
1
answer
531
views
Split sum into equal terms
Given a sum of $l$ integers $r_1+...+r_k+...+r_l$ and an integer $t$.
Find indices
$1 < p_1 <...< p_h <...< p_{t-1} < l$
such that in sum
$(r_1+...+r_{p_1})+...+(r_{p_{h-1}+1}+......
- 11
2
votes
4
answers
2k
views
Efficient algorithm for finding the minima of a piecewise linear function
Consider real numbers $a_i$ and $b_i$ for $i=1\dots n$ and define a function by
$f(x) = \max_i ( a_i + b_i x )$
We desire to find $\min_x f(x)$. Obviously this occurs at an intersection of two lines:...
- 823
6
votes
3
answers
11k
views
Maximum flow with negative capacities?
I'm trying to compute an (s-t) maximum flow through a network which includes a number of arc pairs ((u,v), (v,u)) that have equal, negative capacities (weights). I'm not aware of any efficient ...
- 83
4
votes
4
answers
703
views
efficient way to compute the inversion of the following matrix
Hi, there
I have looked it up in the current textbook. The conventional numerical method to compute the inversion of an $n \times n$ matrix requires $O(n^3)$. However, for the following special ...
- 41
4
votes
1
answer
275
views
Symmetry of the integer gap
Are there results that bound the asymmetry of the duality gap of an integer program? That is to say, if the difference between the LP solution and the IP (primal) solution is $a$, is there a function ...
- 183
0
votes
2
answers
4k
views
Linear programming piecewise linear objective
I am fairly new at linear programming/optimization and am currently working on implementing a linear program that is stated like this:
max $\sum_{i=1}^{k}{p(\vec \alpha \cdot \vec c_i)}$
$s.t. $
$|\...
- 3