Questions tagged [nonlinear-optimization]
Nonlinear objectives, nonlinear constraints, non-convex objective, non-convex feasible region.
287 questions with no upvoted or accepted answers
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solution of an infinite horizon optimization problem
Give the following formulation:
$\min_{\{x_s(t):\forall s,t\}} \sum_{s \in \mathcal{S}} \mathbf{1}\left(\lim_{T\rightarrow \infty} \frac{1}{T} \sum_{t=1}^T \frac{y_s(t)}{x_s(t)}\leq 1\right)$
$s.t. ...
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0
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131
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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
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83
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reconstructing a linear order corrupted by noise
Suppose we have a partial order (efficiently computable), $\leq$, on $\mathbb{R}^n$, a set $S \subset \mathbb{R}^n$, and let $\rho$ be the standard Euclidean metric. We want to find a set $S^\prime = ...
- 479
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237
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Asymptotics to Taylor expansions?
I posted a question on MSE about approximating Taylor series but Despite a bounty I did not receive any answers or comments.
Maybe you guys can help.
https://math.stackexchange.com/questions/1440931/...
- 763
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130
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Optimization question: maximize quadratic objective with semidefinite constraints
I recently encountered the following optimization problem:
$\max \|AX\|_F^2$
subject to: $X\succeq0$ and $Xb_i\leq c_i$ for a collection of $T$ conditions: $i=1,\ldots,T$.
Matrices $A$ and $X$ are ...
- 123
1
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89
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Frobenius nearest non-negative Gram matrix of balanced row-sums
Let $W \in \mathbb{R}^{n \times n}$ be any non-negative real symmetric matrix. For $k \leq n$, let $\mathcal{F} := \{X \in \mathbb{R}^{n \times k} \ | \ X \geq 0, X \mathbf{1} = \alpha \mathbf{1}, \...
- 340
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76
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Minimizing sum of functions, while keeping their values non-negative
Suppose we have data $\mathbf{x}_i$, $i\in \{1, \ldots, K\}$, and we're trying to find parameters $\hat{\mathbf{\theta}}$ such that
$$\hat{\mathbf{\theta}} = \underset{\mathbf{\theta}} {\mathrm{...
- 111
1
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166
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modern exposition of exact ground state of classical XY model or Ising model
What is the state of art technique in solving exact ground state of Heisenberg model, meaning minimization of the H terms (hamiltonian) provided infinite spin space?
http://en.wikipedia.org/wiki/...
- 867
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201
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Optimization of a multilinear function over a product of hypersimplices
Let $P = \Delta_1 \times \cdots \times \Delta_N$ be the Cartesian product of $N$ hypersimplices. Let $f : P \to \mathbb{R}$ be a multilinear function of $N$ variables, ie $x_i \mapsto f(x_1, \ldots, ...
- 171
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49
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Limiting Entropy of deterministic sequences - 2
Consider a collection of increasing positive integers $\{a_i\}_{i=1}^m$ and the distribution $p_i=\frac{a_i}{\sum_{i=1}^ma_i}$. Given $\{a_i\}_{i=1}^m$, let $\mathcal{P}_a$ be limiting distribution $m\...
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100
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Changing a nonlinear equality constraint into some conic inequality plus rank constraint
If we have a constraint optimization problem in which one of our constraint is $\prod\limits_{k = 1}^N {\left( {x - {a_k}} \right) = 0} $ . How could this nonlinear equality condition be changed into ...
- 33
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Estimation of part of parameters from an ODE
Suppose, we have an ODE
$$ \frac{dy}{dt}= f(t,y;p',a)$$
or alternatively
$$ \frac{dy}{dt}= f(t,y;p)$$
where the set of all parameters $p = (p',a)$. We only need to estimate part of parameter set $p'$...
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Subtour Elimination in Travelling Salesman Problem using MTZ
I am trying to formulation a problem similar to a Traveling Salesman with Time Window constraints.
To eliminate subtours, I need to use some constraint similar to a generalization of MTZ constraints ...
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784
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Solving a system of complex non-linear equations
I have a set of five equations which can be described as follows:
$m_{i}=\frac{k_{1}}{(x+a)^{i}} + \frac{k_{2}}{(b+d)^{i}}+ \frac{k_{3}}{c^{i}}$
for i=1 to 5 where
$$\eqalign{
k_{1}&=\frac{a(x+...
- 209
1
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266
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Multiobjective semidefinite programming
Let $C$ be size $n \times n^{2}$.
Let $B$ be size $2^{g(n)} \times n^{2}$ where $g(n) > n$.
There is only one $\mathcal{1}$ per row of $C$ and remaining entries of $C$ are $\mathcal{0}$.
$B$ is ...
- 800
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37
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Maximise norm over the boundary of a convex set
Let $K\subset \mathbb R^2$ be compact, convex and connected. What is the know numerical scheme to find the extremal points of $K$?
Denote by $\partial K$ the collection of all extremal points of $K$. ...
- 1,389
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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}...
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How can we tighten the bounds of the $\ell_1$-norm of $\mathbf{A}x$ where $\mathbf{A}\in\mathbb{Z}^{m\times p}$ and $x \in \{0,1\}^p$?
I am curious about the upper bound of $\|\mathbf{A}x\|_1$ where $\mathbf{A}\in\mathbb{Z}^{m\times p}$ and $x \in \{0,1\}^p$, for a specific $\mathbf{A}$ as defined below.
I know this is an NP-hard ...
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When can a point be reconstructed from relative angle measurements?
Given a set of points $p_1,\dots,p_n$ in $\mathbb{R}^d$ and a target point $x\in\mathbb{R}^d$, I measure all the angles between all pairs of points and the target point. In other words, I have the ...
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35
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A question on the optimisation problem and FWL theorem
Let's say we are considering the following model:
$$
(\beta^{\star},f^{\star}) := \arg\min_{\beta,f \in \mathcal{F}} \mathbb{E}[\left(Z_i - f(X_i, E_i) - \beta^\top \boldsymbol{\tau}_{i,E_i}\right)^2|...
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63
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A maximisation problem : finite or not?
Let $\mathcal M_2$ be the space of real $2\times 2$ matrices and $\mathcal S_2\subset \mathcal M_2$ be its subset consisting of positive semidefinite elements, i.e. $A\in \mathcal S_2$ iff $A$ is ...
- 1,389
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44
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How to formulate piecewise quadratic function optimization without introducing binary variables?
I have a problem with logical constraints (either-or constraints). I know that it can be solved by either big-M or complementary formulations. However, i do not want to convert it into mixed-integer ...
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72
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Minimizing the Spectral Norm of the Hadamard Product of a Quadratic Form Using CVX
I am trying to use CVX to minimize the spectral norm of the Hadamard product of two matrices, one of which is in quadratic form. Specifically, I am trying to minimize $\|{\bf A} \odot {\bf XX}^H\|_2$, ...
- 43
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36
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How to prove the convergence of Gechberg-Saxton algorithm?
I just have a problem that Gerchberg-Saxton algortihm is no worse than the previous iteration
but not sure whether it is convergent.
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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 $...
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1
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104
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Optimality condition for strongly convex function under sparsity constraint
Let $f: \mathbb{R}^p \to \mathbb{R}$ be a $2s$-sparse strongly smooth, $2s$-sparse strongly convex and twice differentiable function. In other words, there exists positive constants $\alpha, L >0$ ...
- 399
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27
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How to control the angles of Kuramoto model by controlling its order parameter?
Consider Homogenous Kuramoto model in this paper. In theorem 3.1, the author derive condition on $A$ such that all second-order critical points of $E(\theta)$ are in two opposite quadrants, by saying ...
- 49
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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
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52
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Seeking help with a matrix optimization problem involving matrix exponentiation
I'm working on an optimization problem where I need to find matrices $P$, $Q$, and $C$ that minimize the norm of the difference between a given matrix $A$ and another matrix defined as $e^{P(Q + Q^T - ...
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153
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Do we have tetration uniqueness by $ A = \inf \sum_n a_n^2 $?
Let $f$ be a real analytic (on at least $|x|<2$) and real solution of the functional equation $f(0) = 1,f(x+1) = \exp(f(x))$. For the existence of such $f$, see here.
Then
$$
f(x) = \sum_n a_n x^n ;...
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129
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Primal optimal attained implies dual optimal attained
Given some optimization problem
$$\min_{x \in S \subset \mathbb{R}^n} f_0(x) \quad \text{s.t.} \quad f_i(x) \leq 0, \quad 1\leq i\leq m$$
we can find the dual problem
$$\max_{\lambda\in\mathbb{R}^m} g(...
- 275
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55
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Relationship of optimal solutions between the total function and the sub function
This is an unconstrained convex optimization problem. Let $\mathcal{N}=\left\{1,\ldots,n\right\}$, $2\leq n<\infty$. Suppose there are many strongly convex functions $f_i(x)$, where $x\in\mathbb{R}^...
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88
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How to solve mckp (multiple-choice knapsack problem) problem with non-linear constraint
How to solve the below optimization problem? $P$ is a probability matrix, $0\le P_{ij}\le 1$. Are there any developed tools to solve this? Thanks a lot.
\begin{equation*}
\begin{aligned}
&\...
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53
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Upper-bound on energy of nonlinear boundary-value problem
The problem:
Consider the following boundary-value problem for the function $\rho : \mathbb{R}^{+} \to \mathbb{R}$ with boundary conditions $\lim_{x\to \infty}\rho(x) \to 1$ and $\lim_{x\to 0}\rho(x)...
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57
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optimization of mixed linear and infinity norm
I have the following optimization problem:
Given a complex sequence $H_i$, $1 \leq i\leq N$. Find a complex sequence $G_i$ that minimizes:
$$ \lambda\cdot\max_i { |H_i\cdot G_i - 1|^2 } + \sum_i |G_i|^...
- 101
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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 ...
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75
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Maximize entropy under Kulback-Leibler divergence
I posed this question in math.stackexchange.com, but have not received any answer. I would like to try my luck here.
In this question, it is to solve
\begin{align}
\max_p &-\int dy\,p(y)\ln p(y) \\...
- 2,239
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156
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Optimal solution of complex optimization problem
Let $Q(x)=a(x)e^{jb(x)}$ be a complex function of $x$. We want to approximate this function with $R(x)=\alpha e^{jx\beta}$ such that
\begin{align}
\text{arg}\min_{\alpha,\beta} \int_{-\frac{A}{2}}^{\...
- 287
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0
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77
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Get a specific number of points from a density distribution area to minimize the average distances
Assuming that an area $A$ on the plane has a known density distribution function $\rho (x, y)\geqslant 0$, now the goal is to obtain $n$ points $p_1, p_2, ..., p_n$ on the area so that $\iint_{}^{} \...
- 143
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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 $\...
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222
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Convergence of ODE solutions almost everywhere to a stable equilibrium point
Theorem: Suppose ${\bf g} :\mathbb{R}^n \mapsto \mathbb{R}^n$ is continuously differentiable, there exists a set $\mathcal{A} \subset \mathbb{R}^n$ such that $\bf g$ is uniformly Lipschitz on $\...
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109
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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|\...
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92
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Optimization problem where the objective function returns a function instead of a real number
As we all know, a classic optimization problem can be represented in the following way:
Given: a function $f: A \rightarrow \mathbb{R}$ from some set $A$ to the real numbers
Sought: an element $x_0 ∈ ...
- 141
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43
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Breaking up an infinite-dimensional optimization problem into a sequence of finite-dimensional problems
My question is a bit vague. I have an infinite-dimensional convex optimization problem and I can solve constrained versions of the problem by restricting the domain of the objective function to a ...
user85365
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46
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Lipschitz solutions to linear complementarity problems (LCP)
Let $M\in\mathbb{R}^{n\times n}$.
For $q\in\mathbb{R}^n$, define the set:
$$S_M(q)=\{y\in\mathbb{R}^n|y\ge 0,q+My\ge 0, y^\top (q+My)=0\}.$$
This is the set of solutions to the LCP $(q,M)$.
We say $...
- 183
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0
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57
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Numerically finding matrix approximation by lower-dimensional "pseudo-similar" matrix
Consider an $N\times N$ (real or complex) matrix $A$, and some $n<N$. Is there a good numerical algorithm that finds the set consisting of an $n\times n$ matrix $B$, an $n\times N$ matrix $I$, and ...
- 3,001
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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 $y(\theta)$ ($\...
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1
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516
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How do I get an analytical solution to this nonlinear equation?
I posted this question over on Math Stack Exchange (link), but have not received a response. I'm wondering if it's too complicated for that audience, so I'm posting it here in the hopes that someone ...
- 109
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0
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36
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Optimizing upper and lower bounds
Let $L_i:X\rightarrow [0,\infty)$ be continuous (objective) functions defined on a metric space $X$ and suppose that
$$
L_1(x)\leq L_2(x)\leq L_3(x)\qquad (\forall x \in X).
$$
Here, I imagine that $...
- 5,405
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0
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52
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An optimization problem about number series
Given $m>0$, we want to minimize
$$
\sum_{k=1}^r a_k \log b_k
$$
for arbitrary increasing number series $a_k\geq 1$ and $b_k\geq 1$ satisfies
$$
\sum_{k=1}^{\infty} \frac{1}{a_k}=1
$$
and $r$ ...
- 1,503