All Questions
Tagged with nonlinear-optimization non-convex-optimization
25 questions
0
votes
0
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77
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Generalizations of Berge's maximum theorem
I have a parameterized optimization problem
\begin{eqnarray}
\max_{x\in D(\theta)} f(x,\theta).
\end{eqnarray}
Assumptions of the standard Berge's maximum theorem are satisfied, so the value function $...
1
vote
0
answers
62
views
Under which condition, such that all second-order critical points satisfy $\sum_j\cos(\theta_i-\theta_j)>0$ for all $i\in[n]$?
Consider the following non-convex function
$$E(\theta):=-\sum_{i,j}A_{ij}\cos(\theta_i-\theta_j)$$
where $A$ is a symmetric, diagonal-free matrix whose non-diagonal element are $\pm 1$. In other words,...
- 49
0
votes
0
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41
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Efficient algorithms to find the global minimum of a non-convex quadratically-constrained quadratic program
I am working on a problem involving a non-convex quadratically-constrained quadratic program and am seeking efficient algorithms to find its global minimum. The problem is structured as follows:
Fix ...
- 1
0
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0
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88
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How to solve mckp (multiple-choice knapsack problem) problem with non-linear constraint
How to solve the below optimization problem? $P$ is a probability matrix, $0\le P_{ij}\le 1$. Are there any developed tools to solve this? Thanks a lot.
\begin{equation*}
\begin{aligned}
&\...
1
vote
0
answers
113
views
Maximisation of a convex (quadratic) function
This post is a continuation of A variant of (discrete) optimal transport problem
For $\alpha=(\alpha_1,\ldots,\alpha_m)\subset\mathbb R^m_+$, $\beta=(\beta_1,\ldots,\beta_n)\subset\mathbb R^n_+$ and $...
- 1,399
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votes
1
answer
139
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An optimization problem with variables on the exponential of a complex number
$$\min_t \quad\operatorname{Re} \sum\limits_{i = 1}^N {\left( {{e^{ - j2\pi {f_i}t}}{r_i}} \right)}, $$where $\operatorname{Re}$ refers to get the real part of a complex number, $\{f_i\}$ is an ...
1
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0
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143
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Can I solve this quadratic program "fast"?
We are given a matrix $D \in \mathbb{Z}^{C \times C}$ of non-negative entries, an integer $k \geq 1$ and we need to maximize the quadratic form $x^T D x$ under some simple constraints. For all ...
- 85
1
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0
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72
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Minimize smooth function $(x,y) \to f(x,y)$ subject to $x \perp y$
Let $V$ be a finite-dimensional real vector space (e.g space of $m \times n$ real matrices equiped with Hilbert-Schmidt inner product $(A,B) \to \mathrm{tr}(AB^\top)$, and let $f:V^2 \to \mathbb R$, $(...
- 6,853
2
votes
0
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448
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Global minimum of sum of a non-convex and convex function, where minima of the non-convex function can be found
I'm interested in finding $\arg\min_{x \in X} (f(x) + \lVert x\rVert_2^2)$ where $X$ is a $[0,1]^n$, $f$ is Lipschitz but non-convex and we already have a procedure to find some $x^* \in \arg\min_{x\...
1
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0
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110
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Minimising kurtosis (non-convex). Can I use algebraic geometry or alternate methods to show uniqueness of a particular solution?
I consider a weighted sum of $n$ identically-distributed correlated random variables. The weights in the sum, $w_i$ for $i=1, 2,...,n$, satisfy $w_i>=0$ and $\sum_{i=1}^{n}w_i=1$. I am ...
- 173
0
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0
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312
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Convergence of heavy-ball method for non-convex optimization
The heavy-ball method (also called gradient descent with momentum) is commonly used in optimization. The update rule can be written as:
$$x_{t+1}=x_t-\eta\nabla f(x_t)-\beta (x_t-x_{t-1})$$
Suppose $\...
- 601
1
vote
1
answer
791
views
Hardness of concave minimization problem
I have an optimization problem $\underset{x}{\min} ~ c(x) - k \cdot x$ where $c(x)$ is a non-decreasing concave function with $c(0) = 0$, $x \in C \subseteq \mathbb{R}^d_{\geq 0}$. By non-decreasing, ...
- 29
6
votes
1
answer
1k
views
Solving a linear program, but over the unit sphere
I want to solve a linear program but with a subset of the variables taken from a unit sphere.
That is, given fixed $\textbf{c} \in \mathbb{R}^{n}$, $\textbf{A} \in \mathbb{R}^{m \times (n+k)}$,
I want ...
- 63
2
votes
0
answers
81
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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
3
votes
2
answers
331
views
Program to solve Optimization Problem
I have an optimization problem, this problem has linear constraints and nonlinear constraints. I solved the linear part by MATLAB but the nonlinear constraints I could not solve it. I downloaded ...
- 31
7
votes
2
answers
569
views
Proving an infinite norm minimization problem has finite support (non-convex p-norms)
Consider an optimization problem over infinite variables:
$$
\begin{align}
\min_{x}~& {\left\lVert{x}\right\rVert }_p
\\
\text{s.t}~& \left\langle x, a_n\right\rangle \ge 1~,~\forall n=1,\...
- 673
4
votes
2
answers
6k
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Maximizing a convex function with a convex constraint
Given a convex function $f : \mathbb{R}^n \to [0,\infty)$, the objective is to find the farthest point in the level set $\left\lbrace x \in \mathbb{R}^n \mid f(x) \leq 1\right\rbrace$ (Assuming that ...
- 151
2
votes
1
answer
105
views
Linear optimization with one positive definite quadratic equality condition in P?
I have the following minimization problem in $z \in \mathbb R^n$, which contains $x_1, \dots, x_t, y \in \mathbb R$.
$$\begin{array}{ll} \text{minimize} & y\\ \text{subject to} & xQx'= y\\ &...
- 13.9k
0
votes
0
answers
181
views
non-convex optimization with constraint
I have a special non-convex optimization problem:
$\min / \max \ f(x) + g(x) + h(x)$,
subject to $| g(x) - h(x)| < \varepsilon$,
where $f(x)$ is non-convex, but both $g(x)$ and $h(x)$ are ...
- 43
1
vote
0
answers
421
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Is this QCQP convex or nonconvex?
\begin{equation}
\begin{split}
\min_{x\in \mathbb{R}^n}\:f(x)=(1/2)x^{T}Q_0x+c_0^T x
\end{split}
\end{equation}
s.t.
$$
g_i(x)=\frac{1}{2}x^T Q_ix-lmax_i\leq0,i\in\{1,...,m/2\}
$$
$$
g_i(x)=\frac{...
- 11
7
votes
2
answers
4k
views
Quadratically constrained linear program (QCLP) over $x$ with the linear constraint $x = Az$
I have a problem that looks very much like a norm-constrained linear program, but with an extra constraint that is unusual for me. The problem is the following. Given a matrix $A$ and a vector $w$,
$$...
- 93
3
votes
1
answer
862
views
Nonconvex optimization problem
I have a nonconvex optimization problem with a linear objective function, a set of linear constraints and a set of nonlinear, non-convex constraints. Is this problem NP-hard? If so, how can I prove ...
- 221
3
votes
1
answer
602
views
Is the feasibility of a system of non-convex quadratic equations and inequations decidable?
I would like to know whether the following problem is decidable.
Given the following system in $x \in [0,1]^n$
$$x^T Q_i x + r_i = 0 \mbox{ for } i = 1, ..., k$$
$$x^T Q_j x + r_j \neq 0 \mbox{ ...
- 165
1
vote
1
answer
655
views
A non-convex quadratically constrained quadratic program
$$\begin{array}{ll} \text{minimize} & \beta^{T} A \beta\\ \text{subject to} & \beta^{T} C \beta=1\\ & \beta \geqslant 0\end{array}$$
where $A, C\in \mathbb{R}^{M\times M}$ and $\beta \in ...
- 13
2
votes
1
answer
1k
views
Can one maximize the spectral norm of a matrix via semidefinite programming?
Consider the following optimization problem:
Maximize $\|X\|_2$, subject to $X$ being Hermitian (or symmetric) and a bunch of semidefinite constraints on $X$. Here, $\|X\|_2$ is the spectral norm of ...
- 1,332