All Questions
Tagged with nonlinear-optimization combinatorial-optimization
23 questions
0
votes
0
answers
22
views
Alignment of unit vectors under graph-neighbor constraints with a global vector
Statement
Let $G = (V, E)$ be a connected, unweighted, and undirected graph with $n$ nodes, represented by its adjacency matrix $A$. Suppose each node $i$ is associated with a unit vector $ \mathbf{v}...
2
votes
1
answer
175
views
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
2
votes
1
answer
274
views
Can we say this nonlinear integer programming problem is NP-hard?
I was wondering if the following nonlinear integer programming problem is NP-hard or not.
$$\max_{x_i \in \{0,1\}} \frac{\sum_{i=1}^{n}a_i x_i}{\sqrt{\sum_{i=1}^{n}b_i x_i}}$$
such that $\sum_{i=1}^{n}...
- 21
1
vote
2
answers
121
views
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
3
votes
1
answer
344
views
How to find the maximum of a sum of squares of sums?
Is there any better than a brute force method for finding the maximum
$$\max\limits_{ (d_{1},\dots,d_{n}) \in \mathbb Z_{m}^{n}} \sum_{j=0}^{m-1} \left(\sum_{i=1}^{n}v_{i,(j+d_{i})\bmod m}\right)^{2}$$...
user494931
0
votes
1
answer
143
views
Transformation of an unconstrained binary quadratic optimization problem into a constrained binary linear programming problem
I know that a constrained linear optimization problem can be transformed into an unconstrained binary quadratic optimization problem (UBQP). Does anyone know if the inverse result is solved in the ...
- 61
0
votes
1
answer
128
views
What is the computational complexity of the calculation of $ \Psi(x) $?
What is the computational complexity of the calculation of $ \Psi(x) $ described below:
Let $\left\{ f_i : \{0,1,\dots,m\} \to \mathbb{R} \right\}_{i=1}^n$. For each $x \in \{0,1,\dots,m\}$ we ...
2
votes
0
answers
70
views
Monotone rearrangements of function, constrained optimization in $\mathcal{L}^p$
By $\mathcal{S}$ let us denote the set of such step functions $f:[0,1]\to [0,1]$ that additionally satisfy:
$$\forall_{ x>\frac{1}{2}} \ \ \lambda\Big(f=x\Big) \ = \ x\cdot \Big[\lambda\Big(f=x\Big)...
user153000
2
votes
0
answers
66
views
Minimizing a certain norm of the identity operator on $\mathbb R^2$
$\newcommand\R{\mathbb R}\newcommand\Q{\mathcal Q}$For mutually orthogonal vectors unit vectors $a=[a_1,\dots,a_n]^T$ and $b=[b_1,\dots,b_n]^T$ in $\R^n=\R^{n\times1}$ (so that $n\ge2$) and for all $x=...
- 128k
5
votes
1
answer
332
views
On a certain norm of the identity operator on $\mathbb R^2$
$\newcommand\R{\mathbb R}\newcommand\Q{\mathcal Q}$For mutually orthogonal vectors unit vectors $a=[a_1,\dots,a_n]^T$ and $b=[b_1,\dots,b_n]^T$ in $\R^n=\R^{n\times1}$ (so that $n\ge2$) and for all $x=...
- 128k
2
votes
0
answers
101
views
Sparse signal recovery (nonlinear case)
Let $K \subset \mathbb{R}^n$, it may be that $K$ is "very thin" (e.g. $K$ is a $k$-dimensional affine subset of $\mathbb{R}^n$, with $k \ll n$). I'm interested in the case where $K$ is ...
- 981
2
votes
0
answers
81
views
Solving Mixed-Integer Non-Linear Optimization Problem
I would like to solve the following optimization problem:
\begin{array}{ll}
\underset{x_{i}\geq0,\, \pi_{i}\in\{0,1\}}{\text{minimize}} & \displaystyle\sum_{i=1}^n x_i\\
\text{subject to} & ...
- 21
1
vote
0
answers
190
views
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
2
votes
1
answer
107
views
Does this simple non-convex problem involving discrete phase shifts have an exact solution?
Let the optimization problem be
\begin{equation}
\max_{\phi_n} \left|\sum_{n=1}^N e^{i\phi_n} a_n \right|,
\end{equation}
where $a_n\in\mathbb{C}$ and the optimization variables have discrete phase ...
- 345
2
votes
1
answer
188
views
$0/1$ programming multiple quadratic constraints
If we have an $n$-variable rank $n$-linear system it is clear we can find whether there exists a $0/1$ solution in polynomial time.
If we have an $n$-variable degree $2$ system how many constraints ...
- 13.9k
6
votes
1
answer
5k
views
max-min optimization problem
I'm curious if there is any nice way to approach solving the following kind of optimization problem. Given a $n \times m$ matrix $A = (a_{ij})$, I want to solve
\begin{align*}
& \max_{c}\min_{1 \...
- 161
1
vote
0
answers
131
views
Lower bound on the value $\textbf{1}^Tx$ such as $Ax\geq b$
The problem may be formulated as follows:
We are given a set of $m$ positive numbers $\{b_1,...,b_m\}$ and a set of $n$ positive numbers $\{v_1,...,v_n\}$. We have $v_j\leq K$, $j=1,...,n$, for a ...
- 123
0
votes
0
answers
47
views
Complexity of optimizing a bi-objective function with integer constraints
I have two different objective functions, each of which can be solved optimally in polynomial time. Does this mean, I can optimize a linear combination of these objective functions in polynomial time ...
- 121
2
votes
1
answer
199
views
Optimization over symmetric polynomials
Consider the following constraint satisfaction problem: Let $\alpha_1 , \ldots, \alpha_k \in \mathbb{R}$ be given as well as an error parameter $\epsilon$. Find $p_1, \ldots, p_n$ such that
(i) $0 \...
- 563
4
votes
0
answers
522
views
Solution of a linearly constrained quadratic programming problem [closed]
What is the solution of the following optimization problem:
\begin{align}
&\min{\mathbf{p}^\mathrm{T} \mathbf{B} \mathbf{p}}\\
&\text{subject to}: \mathbf{0}\leq{\mathbf{p}}\leq \mathbf{1}.
\...
- 51
1
vote
1
answer
370
views
Maximizing a certain concave function over a non-convex set
I have been working on a problem which involves maximizing a concave function over a non convex constraint set. The problem is the following $$max. \frac{1}{2}x^TBx\\ s.t.\ x_i(1-x_i)=0,\ \ i=1,\cdots,...
4
votes
1
answer
396
views
Optimization problem - maximizing number of satisfied linear inequalities subject to a quadratic constraint
I am wondering what is known about optimization problems of the following type.
Our control x is a unit vector in $\mathbb{R}^n$. We are given a finite number of linear inequalities
$$Az≥b,$$
and we ...
- 185
5
votes
1
answer
3k
views
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 ...
