All Questions
Tagged with combinatorial-optimization convex-optimization
23 questions
1
vote
0
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29
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Integral hull of a polyhedron Q is polyhedron
Let $Q \subseteq R^n$ be a rational polyhedron and let $Q_I=Convexhull(Q \cap Z^n)$. By finite basis theorem, we have $Q=P+C$ for some rational polytope $P$ and finitely generated cone $C$ where $C=R_+...
- 21
0
votes
1
answer
236
views
Solving a 0-1 quadratic matrix inequality
I am working on a binary optimization problem. So far I have derived the following constraint functions.
\begin{align}
\begin{bmatrix} \left( P + \sum_{i=1}^n (\sum_{j=1}^n x_{i, j} \alpha_j) e_i e_i^...
- 11
3
votes
2
answers
215
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Is there a closed-form solution for $\max_D \operatorname{Tr}(ADBD)$
Is there a closed-form solution for
$$\max_D \operatorname{Tr}(ADBD)$$
where $D$ is a $N\times N$ diagonal matrix with $m<N$ number of $1$'s and the rest are $0$'s, and $A$ and $B$ are real ...
- 433
2
votes
1
answer
175
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Optimization over permutation
The Problem
This is the problem I am working on: Given a set $X = \{x_1, x_2, \cdots , x_n\}$ in a metric space, find an optimal ordering $\pi : X \rightarrow X$ that maximizes the following objective ...
- 43
1
vote
2
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121
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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 $\mathbf{x}_i\in\mathbb{R}^d$.
...
- 175
2
votes
0
answers
56
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A variant of the elliptope relaxation
Given a p.s.d. matrix $A$, one may want to find:
$$
\max_x x^t A x \mbox{ such that } x \mbox{ has entries }1 \mbox{ or } {-1}.
$$
This hard problem has a well known relaxation based on the so called ...
- 2,772
0
votes
1
answer
214
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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
6
votes
1
answer
176
views
Subsets of a ball/sphere with the largest sum of distances
$\newcommand\R{\mathbb R}\newcommand\S{\mathbb S}$Let $B_d$ and $S_{d-1}$ denote, respectively, the closed unit ball and the unit sphere in $\R^d$. Let us say that a finite subset $F$ of $B_d$ is ...
- 128k
1
vote
1
answer
94
views
Weak submodularity for consecutive indices
Let $f\colon \mathbf{R} \times \mathbf{R}^+ \rightarrow \mathbf{R}$ be defined by $f(x,y) = \frac{x^2}{y}$. Let $X = \left\lbrace x_1, \dots, x_n\right\rbrace \subseteq \mathbf{R}$, $Y = \left\lbrace ...
4
votes
1
answer
338
views
Minimization of a discrete valued function
$$
\min_{f} \sum_{i=1}^n \max \left( 0, 2f(i) - f(i-1) -f(i+1)\right),
$$
where the minimum is taken over all the functions $f$ from $\{0,1,2,\ldots,n+1\}$ to $[0,x]$, $x <1$, such that $f$ is non-...
user83947
2
votes
1
answer
240
views
Basis pursuit algorithms for exponentially large matrices?
Are there any efficient algorithms/heuristics for basis pursuit for exponentially large matrices?
That is
$$\begin{array}{ll} \underset{x \in \Bbb R^n}{\text{minimize}} & \lVert x \rVert_0\\ \text{...
- 21
0
votes
1
answer
61
views
Variant of the linear programming problem
Good afternoon, my experience in mathematical programming is low. I would like to know if there is any general method to address the following problem:
$$\text{Minimize }\sum_{i=1}^n d_i(x_j)$$
$$s.a....
- 193
5
votes
2
answers
633
views
Exactness of the semidefinite programming (SDP) relaxation of maximum cut (Max-Cut)
Currently, what conditions are known to be sufficient for the SDP relaxation of Max-Cut to be exact?
- 51
5
votes
1
answer
432
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Optimizing a multivariate symmetric (permutation-invariant) function
Let $\ell$ and $d$ be two integers such that $\ell \le d$.
I would like to find the global maxima of the following symmetric function $f\colon (0,1]^n \to \mathbb{R}$,
$$f(x_1, \ldots, x_n) := \sum_{\...
- 381
0
votes
0
answers
72
views
Discrete primal-duality in optimization
I would like to inquire about the existence of something perhaps similar to the duality theorem in the convex analysis or convex programming in the discrete setting. Here is a concrete example.
Let $...
- 2,239
4
votes
0
answers
539
views
Using Linear Programming as an iterative procedure
Suppose, we have a linear program and an optimal solution to it. Suppose now, we get a new constraint. We want to obtain an optimal solution to the given linear program extended by that new constraint....
- 391
0
votes
0
answers
68
views
Under which conditions discrete versions of convex\concave function are submodular/supermodular?
I have $f(x)$ with $x \in [0,1]$ and $f(x)$ is convex, then, under which conditions discrete function which is defined as $f(x_h)$ on discrete subset of $[0,1]$, for example, $x_h \in \{0, h, 2h, \...
- 355
4
votes
0
answers
108
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Designing Character Other Than Temperature for Simulated Annealing on Combinatorial Optimization
Many research on designing temperature for simulated annealing is carried out. We wonder if there is any research on designing general feature of the Hamiltonian used in Simulated Annealing.
For ...
- 909
3
votes
1
answer
239
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A difficult combinatorial optimization problem
Let $\mathcal{J}$ be a closed, bounded, compact, convex set in $\mathbb{R}^L$.
(Notations: vector $\mathbf{x}$ is denoted in bold letters and its $i^{th}$ co-ordinate is denoted as $x_i$. $\mid\...
- 1,421
0
votes
0
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113
views
Sufficient optimality condition for a non-smooth quasiconvex problem
The result of relaxing to an integer program is the following optimization problem:
$$\min_{\textbf{x}} \sum_{i=1}^n \alpha_i h(x_i)\quad subject \; to \quad A\textbf{x} = \textbf{0}$$
where $\textbf{...
- 127
2
votes
2
answers
798
views
Survey on Compared Running Time: Ellipsoid Method vs. Simplex Method
If you look through papers on the Ellipsoid Method, there is a large agreement, that the Ellipsoid Method, although theoretically polynomial, is in practice way slower than the Simplex Method. ...
- 329
5
votes
1
answer
3k
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Maximizing supermodular functions
I have a real supermodular objective function which I want to maximize with constraint. The constraint is on the size, like |A|=k .
I am wondering if anyone can give me more information about a ...
3
votes
1
answer
886
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Is the max of two supermodular functions supermodular?
A function $f: \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ is supermodular if for every $x'>x$ and $y'>y$,
$$f(x',y') + f(x,y) > f(x',y) + f(x,y').$$
Suppose $f$ and $g$ are supermodular, ...
- 31