All Questions
Tagged with integer-partitions optimization
18
questions
0
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56
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A generalized subset sum problem
I'm looking at a problem I believe is combinatorial.
Find all possible solutions $\mathbf{x}$ to:
$$\mathbf{a} = [a_1, a_2, ..., a_n], a_k \in \mathbb{N^+}$$
$$\mathbf{l} = [l_1, l_2, ..., l_n], l_k \...
- 231
1
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0
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121
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Minimizing the magnitude of the sum of a vector of complex numbers with an integer constraint
Let $h_{1}, ..., h_{N} \in \mathbb{C} $
Consider minimizing the function below:
$ \min \left| \sum\limits_{i=1}^N h_{i} x_{i} \right| $
with the constraints $x_{i}^2 = 1$
i.e., $x_{i}$ can only take ...
- 11
3
votes
1
answer
79
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Sum of integers closest to a given number
I have the following problem at hand:
Given an odd number $n > 7$, find a set of non-negative integers $m_7$, $m_8$, ..., $m_{13}$ and $m_{14}$, such that the sum
$m_7\cdot 7 + m_8\cdot 8 + ... + ...
- 41
2
votes
0
answers
214
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Maximize sum of ceiling functions
I need to find the maximum of a sum of ceiling functions. The following are given
$$N,C\in\mathbb{Z}\text{ with }0\leq N\text{ and }1\leq C$$
$$\frac{p}{q}\in\mathbb{Q}\text{ with }p,q \text{ coprime ...
- 87
5
votes
2
answers
412
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Maximizing product of three integers
It is well-known that, if we want to partition a positive number $m$ into a sum of two numbers such that their product is maximum, then the optimal partition is $m/2, ~ m/2$. If the parts must be ...
- 10.8k
3
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52
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How to find smallest sum of denominators so that sum of fractions is less than or equal to 1, given the numerators? [closed]
Given $n$ positive integer numbers $a_1, a_2, \dots, a_n$, how do I find $n$ positive integer numbers $b_1, b_2, \dots, b_n$, so that
$$\frac{a_1}{b_1} + \frac{a_2}{b_2} + \dots + \frac{a_n}{b_n} \leq ...
- 131
1
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44
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Partition an integer $n$ into parts so as to maximize the product of the parts [duplicate]
We are given an integer $n\geq 3$. Our goal is to partition $n$ into $k$ parts $p_1, p_2, \ldots, p_k$ with $p_1 + p_2 + \ldots + p_k = n$ (for an arbitrary $k$ we can choose) so as to maximize the ...
- 281
1
vote
1
answer
30
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Searching for the right partitions
I have a set of integers $U$ with cardinality $311$ and whose sum is 500 dollars.
I want to group all the elements of this set into three different subsets:
The sum of the elements of subset $A$ is $...
1
vote
0
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88
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Maximum value, function of partition and its conjugate
Suppose that we have $n\in \mathbb{Z}_{+}$ and some $\alpha\ge 3$.
I am trying to find maximum value of:
$\sum_{i,j=1}^{n}|\lambda_{i}-\lambda_{j}^{*}|^{\alpha},$
over
$\{\lambda\in \mathbb{Z}^...
user617199
1
vote
0
answers
46
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similar to landau's function, but all elements are odd
Everyone knows about landau's function, then I was curious about the lcm of partition "n" which element's are all odd numbers.
for example:
$$8=3+5 \Rightarrow f(8)=15$$
$$19=3+5+11 \Rightarrow f(...
- 11
0
votes
0
answers
421
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Partition a number $n$ in exactly sum of $k$ distinct numbers such that product of the numbers should be maximum.
The question is to partition a given number $n$ in exactly sum of $k$ distinct positive numbers such that the product of $k$ distinct number become maximum. $k$ will be given optimally so that it will ...
0
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0
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42
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Upper bound of number of k-tuples whose components are non-negative integers and sum up to n?
Is there any nontrivial upper bound for the size of $\{(x_1,\dots,x_k)\in\mathbb{N}^k:\sum_{i=1}^kx_i=n\}?$ We may assume $k,n$ are large enough but $k\ll n$, so an asymptotic bound is also helpful.
- 2,085
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votes
1
answer
79
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discrete convexity arising in a simple discrete optimization problem
Let $S$ be a fixed integer satisfying $S \ge 1$, let $a$ range over the integers between $1$ and $S$ inclusive, and for $i = 1, \dotsc, a$, let each $x_i$ range over the nonnegative integers, such ...
1
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0
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27
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Use of the "Optimality Principle" in a problem regarding maximization of a product indexed by a partition of a natural number.
I am currently reading a paper (Iterated Binomial Coefficients by S.W. Golomb, The American Mathematical Monthly, 1980, 719-727) that makes use of the "optimality principle" in a couple of proofs. One ...
- 3,423
0
votes
1
answer
285
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Maximizing the sum of exponentials whose exponents sum to $N$
Let $N \geq 1$ be a sufficiently large integer, let $a > 1$ be a real number, and let $n_1, \dots, n_t$ be integers between $0$ and $K$, where $K$ divides $N$. I want to determine the following:
$$...
- 3,600