All Questions
213 questions with no upvoted or accepted answers
1
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95
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Existence of polytope
Does there exist a polytope in dimensional d consisting of $k>d+1$ faces satisfying that every d faces intersect? I tried 3 dimensional cases, and it seems negative. But is it all negative for any ...
- 43
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
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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
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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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0
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94
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Does the set of points minimizing their distance to a multiset of convex polytopes result in a polytope?
Let $\mathbb{R}^n$ be a normed affine space of finite dimension $n$, and $d : \mathbb{R}^n \times \mathbb{R}^n \mapsto \mathbb{R}^+$ be the distance derived from the norm under consideration. A convex ...
- 125
1
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0
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75
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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
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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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196
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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
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256
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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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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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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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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
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
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0
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21
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Easy instance of set cover
I am trying to prove that a natural greedy algorithm solves the following instance of the set cover problem: for a set of elements $e\in U$ with a set of weights $w_e$, we define the cost of a subset ...
- 4,049
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0
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48
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A question on a quantitative form of Farkas' lemma
Suppose A is an $m \times n$ matrix whose entries are non-negative integers and $\mathbf{b}$ is a vector with rational entries. A version of Farkas lemma implies that if the equation $$A\mathbf{x}=\...
- 6,214
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115
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Software for computing polytopes
As can be inferred from the title, I want to do some computation on the facets representation of the polytopes given the vertices. My advisor recommended me Polymake, which is indeed useful even with ...
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0
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52
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Polyhedra volume, faces and edges from vertices
Given a set of vertices in 3D corresponding to a convex polyhedron, what is the most efficient way to find its volume, faces, and edges?
I've found some techniques using convex hulls. But I think I ...
- 101
0
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0
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39
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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 ...
- 101
0
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0
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30
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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 $\alpha_2 > \beta_2 ...
- 23
0
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0
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36
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ILPs with square constraint matrix
Given the Integer Linear Programming ($\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,\\ &...
- 13.2k
0
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0
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26
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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 ...
- 13.2k
0
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0
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171
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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 ...
0
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64
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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 ...
- 3
0
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0
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164
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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.
...
- 129
0
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1
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114
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Mixed integer program and continuous Diophantine approximation
Let $n\in\mathbb{N}$ such that $n\geq 2$ and let $0<r<1$ be a real number. We wish to solve the following problem.
$$\min_{(t,(z_j)_{j=2}^n) \in \mathbb{R}\times \mathbb{Z}^{n-1}} t$$
subject to ...
0
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0
answers
55
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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}^...
- 1
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0
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85
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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} [ ...
- 671
0
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0
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145
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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\|_{...
- 439
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0
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84
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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(...
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272
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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}=...
- 4,058
0
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0
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94
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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 ...
- 21
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1
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397
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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 ...
- 1
0
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0
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115
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Explicit equation for border of the Minkowski sum of sets
Assume we have sets of the form
$$
M_j = \{x\in\mathbb{R}^d : f_j(x) \le 0,x \ge 0\}
$$
where $x\ge 0$ means $x_i \ge 0 \quad \forall i=1,\dots, d$.
Goal
I am looking for an (explicit) representation ...
- 385
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0
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124
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The best unitary matrices that approximate a matrix product
Let $\mathbf{A}$ be an arbitrary $N\times N$ complex matrix. Moreover, $\mathcal{U}_1$ and $\mathcal{U}_2$ are distinct subsets of all unitary matrices. Suppose the matrices $\mathbf{U}_1$ and $\...
- 287
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0
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85
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Realizability of abstract polytopes
What are the conditions that allow an abstract polytope to have a non-skew (but not necessarily convex) realization in Euclidean space?
- 2,773
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0
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109
views
How to find a set given its support function
Let $\mathcal{U}$ be a convex and compact set. Its support function is defined as $\delta^*(v|\mathcal{U})=\sup_{u\in \mathcal{U}} v^T u$. Assume that we are given the support function $\delta^*(v|\...
- 53
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40
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Subtour-gluing constraints for ILP formulation of TSPs
If one doesn't want to introduce additional variables to the ILP of a TSP instance, one has to add exponentially many so-called subtour-elimination constraints; in practical calculations subtour-...
- 13.2k
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96
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Why is Gaussian distribution always chosen for smoothed analysis?
I came across the algorithmic perfomance analysis model of smoothed analysis. In all references that I read a Gaussian distribution was used for perturbation (e.g. Spielman and Teng 2004 for the ...
- 29
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97
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Possible new convex uniform polytope
Does there exist a convex uniform 9-polytope obtained by diminishing the 9-hypercube, removing 480 of its 512 vertices and turning each 8-hypercube facet into an 8-orthoplex?
- 2,773
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165
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Minimum circumscribed ellipsoid of $\mathcal H$-polytope
Given matrix $A \in \mathbb{R}^{m \times n}$ and vector $b \in \mathbb{R}^n$, consider the $\mathcal H$-polytope $P$ defined as follows
$$ P := \left\{ x \in \mathbb{R}^n : Ax \leq b \right\} $$
I ...
- 235
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0
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137
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Any technique for linearization, or linear approximation?
Consider the following Matrix constraint:
$$
\begin{bmatrix} -U+\psi\Sigma_b^{-1} & V \\ V^T & -V^TU^{-1}V+\tau_2 -\psi \end{bmatrix} \leq 0
$$
where $\Sigma_b$ is a known positive definite ...
0
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0
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72
views
Regiment map from Coxeter-Dynkin diagram
The following problem arises in the attempt to enumerate uniform polytopes:
Given a Wythoffian polytope, what are all other Wythoffian polytopes which use maximal subsets of its vertices and edges, ...
- 2,773
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93
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Number of vertices in a polyhedron
Consider polytopes
$$A_1[x_{1,1},\dots,x_{1,m_1},z_{1}]'\leq b_1$$
$$A_2[x_{2,1},\dots,x_{2,m_2},z_{2}]'\leq b_2$$
$$B[z_{1},z_{2},z]'\leq c$$
having vertex count $v_1,v_2$ and $v$ respectively.
We ...
- 13.9k
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0
answers
68
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Convex optimization under asymmetric loss in infinite dimensional space
The following problem is common in financial economics
$$ \min_{m \in L^2} \mathbb{E}[ \phi(y(\theta)-m)] \quad \text{s.t. } \mathbb{E}[ mx ]= q $$
That is, given a random variable $y(\theta)$ ($\...
- 45
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0
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108
views
Solutions to matrix equations in the non-negative integers
For an integer matrix $S$, and an integer vector $y$, I'm looking for solutions to $xS = y$ where the entries in $x$ are in the non-negative integers.
I've been doing this with Sage's mixed integer ...
- 101
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0
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43
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Minimizing along independent directions, nonlinear programming
Good afternoon, I am studying the book Nonlinear Programming: Theory and Algorithms (by Mokhtar S. Bazaraa, Hanif D. Sherali, C. M.) particularly the Theorem $7.3.5$. I'm not sure I understand this ...
- 193
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0
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50
views
Every point in a regular polytope has its own antipodal point or antipodal face
I apologize for using non-common language. When this problem comes to my mind, it seems quite easy but It's not.
Maybe It can be rewritten as,
There exists a unique facet containing the most far ...
- 285
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votes
0
answers
101
views
How can we analytically solve this max-sum-min problem?
Let $I$ be a finite set, and $A_{ij},B_{ij},x_i,y_j\ge0$. I want to find the choice of $x_i,y_j$ maximizing $$\sum_{i\in I}\sum_{j\in J}A_{ij}\min\left(x_i,B_{ij}y_j\right)\tag1$$ subject to $$\sum_{i\...
- 167
0
votes
0
answers
35
views
Converting a vector in a cone statement to inequality constraints
I would like to convert the following condition for $x$
\begin{align}
x = N \lambda, \lambda \geq 0
\end{align}
to a pure linear inequality form, i.e. find an $L$ and eliminate $\lambda$
\begin{...
- 1
0
votes
0
answers
232
views
What do square roots as minimums have to do with Harmonic numbers?
In an earlier question where I conjectured (and GH from MO confirmed) that the von Mangoldt function is the limit at s=1 of a certain Dirichlet series:
$$\Lambda(m)=\lim_{s\to 1+}\zeta(s)\sum_{d\mid ...
- 1,183