Questions tagged [constraints]
For questions about computationally solving a problem subject to some constraints on the solution. For optimizations, apply the more specific [constrained-optimization] tag instead.
56 questions
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Solve Large Scale Underdetermined Linear Equation with per Element Equality Constraint
I have the following system on $\boldsymbol{x}$:
$$ \boldsymbol{A} \boldsymbol{x} = \boldsymbol{0}, \quad \text{subject to} \; {x}_{i} = {v}_{i} \; \forall i \in \mathcal{V} $$
Where $\boldsymbol{A} \...
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1
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How to add damped constraint force to constrained dynamics simulation?
I have implemented a constraint dynamics physics simulation as proposed by Andrew Witkin et al 1990, but I cannot get the initial constraint "snapping" correctly.
I implemented
$$ JWJ^{T} \...
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Solving linear system of equations with constraints on unknowns
I would like to solve a system of linear equations $y=Uh$ for an unknown vector $h$, where I have a few constraints on some of the elements of $h$. The matrix $U$ is composed of a vector $u$ (length $...
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Order of Error - Confusion: Clarifying Constraints on Constants and Determining Order of Error
I'm struggling to determine the order of error when considering the error value denoted by $\text{err}$ in relation to the variable $h$. Specifically, I aim to ascertain the value of $x$ in the ...
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220
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Damping not working in Verlet simulation with absolute constraint
I'm trying to build a simple physic simulation with Verlet intergration, which I'm currently implementing as such:
...
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2
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1
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Solving constrained odes's using inbuilt solvers in Matlab/Octave
I would like to solve a set of coupled second order differential equations using inbuilt Matlab/Octave subroutines. These equations arise when trying to model sliding of mass ($m_2$) over a wedge of ...
5
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161
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What are the various methods in adding an additional constraint to the quadratic spline interpolation problem
I am taking a class on numerical analysis. While the professor was deriving the theory behind quadratic splines, the professor mentioned that a quadratic spline function has the form:
$$
p_{i}(x)=a_{i}...
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349
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SLSQP solver scipy with linear subset constraints
I have been trying to solve a least squares problem of the following form:
$$
\begin{equation}
\min_{\vec{x}} \frac{1}{2} \lVert f(\vec{x}) - f_{\text{target}} \rVert_{2}^2 + \alpha\Big( \frac{1-\rho}{...
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92
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How "kinematic" rigid bodies are implemented in physics engines
In most physics engines there's this concept of "static" bodies, which act as rigid bodies with infinite mass. Then there are "kinematic" bodies that act as static bodies, but ...
- 143
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49
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The condtion for Augmented Langrangian Multiplier
I am currently learning the usage of Augmented Lagrangian Multiplier to achieve my equality constraint. I have learnt from the https://en.wikipedia.org/wiki/Augmented_Lagrangian_method that I have two ...
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Hanging nodes in deal.ii tutorials: how is the continuity constraint imposed?
While looking at step6 of deal.ii tutorials, I decided to try to understand how the constraints coming from hanging nodes are imposed. So I started by watching video lecture 16 by prof. Bangerth
As ...
- 435
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Differences in using Clausius-Duhem inequality vs Principle of Virtual Work/Power in derriving constitutive equations?
I am a novice getting my toes wet in continuum mechanics and nonlinear elasticity. I have seen papers that use both approaches in developing constitutive connections to compliment balance equations of ...
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563
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How to best code a problem with scipy, cvxpy or Convex.jl with given generated data
I have a curve fitting problem of the form:
$$
\textbf{y} = f(\textbf{x}, a,b,c,d) + \varepsilon
$$
$$
f(x, a,b,c,d) = \frac{b}{e^{x\cdot a}+c}+d
$$
with the constraint
\begin{equation}
\begin{aligned}...
- 141
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Is there a way to bound the values of a variable when using scipy.integrate.solve_ivp in python?
I want to solve an IVP in python with two variables, x and u, but I need the values of u to be between 0 and 1. Right now it is giving me a solution with negative values for u. Here is the code I have....
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Constraint programming problem with conditional constraints and some unknown indicator variables
I have an interesting little problem that I believe can be formulated in terms of optimization or constraint programming. I have a few dozen variables $a$, $b$, $c$ ... and a set of constraints that ...
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Setting up optimization problem in GEKKO
I have the following dynamical system,
$\frac{d \phi}{dt} = -M^TDM\phi \tag{1}\label{1}$
$\frac{d \hat\phi}{dt} = -M^T\tilde{D}M\hat \phi \tag{2} \label{2}$
$\eqref{1}$ represents the exact ...
- 433
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130
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Constraining the total volume in Finite Element Methods
I have a diffusion problem which can be broken down to be:
$-\Delta u = f(u) $ on $\Omega ~/~ \Omega_{int}$
$u = 1$ on $\Omega_{int}$
Note that this is an internal Dirichlet constraint to the ...
- 3,065
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1
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1k
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sum of absolute difference constraint in optimization problem
I am writing a model for an optimization problem. I need to write the following constraint:
$$\sum^{N - 1}_i \lvert (a_i - a_{i+1}) \rvert \leq 2\, .$$
How to write this constraint (or linearize)?
...
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33
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Semi-Definite relaxation of non-linear constraint?
I am implementing an optimization problem using semi-definite approach. One of my constraints is of following form
$ trace(A∗X)−(k∗trace(A∗X))+(k∗\sqrt {(trace(B∗X)} )==0$
where k is a constant, A ...
2
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0
answers
723
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What is the mathematical and physical principle behind of RBE2 element？
I am writing a 3d linear finite element code to solve the standard linear elasticity equation on a tetrahedron mesh of a gearbox. Notice that, the two rectangular plates above the gearbox are fixed, ...
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Domain for convex perspective function
The perspective of a function $f : \mathbb{R}^n \to \mathbb{R}$ is the function $g: \mathbb{R}^n \times \mathbb{R} \to \mathbb{R}$ where $g$ is defined as $$g(x,t) = tf(x/t)$$ with $$\mathbf{\text{dom}...
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Constraints 'exactly/at most one non-zero element' without binary variables
In a much larger MINLP problem, I have set of variables $\{a_{ij}\}_{m,n}$, such that $0 \leq a_{ij} \leq 1 $ for all $i$, $j$, which I could think of as a matrix, for which I have two requirements:
...
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Defining a soft constraint in cvxpy
I am using cvxpy to do a simple portfolio optimization.
I implemented the following dummy code
...
- 111
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2
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160
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Making difference of log constraints convex
I have the discrete likelihood estimation problem $\max \sum m_i\log p_i $ where $m$ is a given vector of length $n$. The constraints are $0 \preceq p \preceq 1$, $\sum_{i=1}^n p_i = 1, $ and one ...
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Linear constraints for L-BFGS-B
I know L-BFGS-B only supports simple box constrains of the form: $l_i \leq x_i \leq u_i$, where $l_i$ and $u_i$ are constants. For my specific optimization problem, I need to specify some simple ...
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2k
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Constrained simulated annealing
Simulated annealing is a useful technique for finding near-optimal solutions to combinatorial problems. I have found a lot of tutorials on implementing the basic algorithm, but miss a general guide as ...
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860
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Trajectory optimization for smoothness
I want to achieve the following in 2D (and without obstacles):
Given start position A and end position B, generate the path between the two points that optimizes a cost function that depends on total ...
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341
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the augmented global stiffness matrix is not positive semi-definite using Lagrange Multipliers method within FEM
The augmented global stiffness matrix is not positive semi-definite when using Lagrange Multipliers method to enforce boundary constraints on a simple square domain of integral form:
I am considering ...
- 321
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Newton's method with box-constraints
I have to use an iterative method (Newton-Raphson, modified Newton and Broyden) to solve a system of nonlinear equations $f(x)=0$. Every unknown $x_i$ is bounded between $l_i$ and $u_i$, i.e., $l_i<...
- 459
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228
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Eigenvalue problem constrained with a penalty method
I am trying to constrain an eigenvalue problem. I am aware of the method utilizing the nullspace of the constraint vectors but I was wondering if it would be to use a penalty method for the same ...
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4
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Simple methods for solving 2D steady incompressible flow?
I'm trying to make a CFD model where I can place a source and a sink anywhere in a grid and get the fluid flow rate across each cell boundary between those locations. I'm starting simple with a 3x3 ...
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3
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648
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How to handle the quadratic constraint $x y \leq z$?
I have a quadratic constraint of the form
$$ x_1 x_2 \leq x_3 $$
where $x_1, x_2, x_3 \geq 0$. The objective is linear, and all other constraints are linear, too. I know that I can transform the ...
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864
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Solve integral equation for unknown constant
Consider the equations
$$\int_0^L \mathbf W(\mathbf u, s) \, \mathrm ds = \mathbf 0$$
where $0 \leq s \leq L$ and $\mathbf u$ is a vector of constants.
Numerically, what is the best way to ...
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468
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FEM, Direct Stiffness Method with a nonlinear displacement constraint in one node
i have a question about a FE problem im working on.
I made a finite element model of an linear elastic block of material (double striped block) attached with a rigid connection to the environment (...
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Projection on Stiefel manifold after integration step
A few days ago, I asked how constraints like $A^T A = I$ can be implemented if one wishes to integrate differential equations of the form $\dot{A}=f(A,t)$. Kirill was so kind to point out that a ...
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Markov (Chain) image generators?
Markov Chains can be used to generate, or auto-complete, text.
https://en.wikipedia.org/wiki/Markov_chain#Markov_text_generators
Training text is read, and some information about the text is ...
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226
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Integration of differential equation with orthogonality constraint
Lets say I have a system of differential equations which has the form
$$\dot{C}_{\alpha,\beta,m} = f_{\alpha,\beta,m}(C_{\alpha,\beta,1},\ldots,C_{\alpha,\beta,N};t).$$
The $f$s are some functions of ...
- 277
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2
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493
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Prescribe solution of a PDE at specific points
I am using MATLAB's PDE toolbox to solve the differential equation
$-\nabla\cdot\left(c(x)\nabla u(x)\right) + a(x)u(x) = f(x)$
The particular problem in question is an electrostatic problem, but ...
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Maxwellian distribution of velocities with Shake algorithm present
I am writing a code to perform hybrid monte carlo molecular dynamics. To do this, I need to have a code to initialize the velocities of all particles according to a maxwell distribution. The code is ...
- 291
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Minimization constraints without using Lagrange Multiplier
Currently, I am working on an unconstrained energy minimization function, but I need to add some constraints. My system is a 2D lattice with a force applied to it, and I want the sides to be able to ...
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0
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55
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Numerical Implementation of "integrates to some values" type constraint in convex solvers?
I am maximizing a linear functional subject to an integrates to one constraint. More explicitly, my problem is
$$\begin{align}
&\max_{x \in \mathbb{R}^n}\quad c \cdot x\\
&\text{subject to} \...
- 486
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Convex Polygon Intersection
Determining the intersection of two convex polygons is one of the fundamental problems in computational geometry .
I'm asking for an algorithm having:
INPUT:
Given two convex polygons P and Q in 2D (...
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2
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141
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How does constraint resolution affect the stability/accuracy of numerical integration?
I understand some basic analysis techniques (local truncation error, global error, zero-stable, absolute stable, etc.) of numerical integration.
But I find it hard to apply these techniques in ...
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508
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Transform from constrained to unconstrained optimization
In a constrained optimization problem, I found in a paper a way to define new variables such that the constraints disappear. They only give the new variable definitions, and I would like to understand ...
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About Convex Geometry
A consistency notion in constraint programming:
Let $P = (X, D, C)$ be a CSP.
Given a set of variables $Y \subseteq X$ with $|Y| = k -1$, a locally consistent instantiation $I$ on $Y$ is $k$-...
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solving a linearly-constrained sparse linear least-squares problem
[ question reposted from https://math.stackexchange.com/questions/786612/solving-a-linearly-constrained-sparse-linear-least-squares-problem ]
Given the system of equations
$Ax=b$, subject to $Cx\le ...
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2
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212
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How to impose a constant constraint PDE
What is the best way to impose a "constant constraint" for a PDE? Specifically, I want to solve an eigenvalue problem $Au=\lambda u$ on the rectangle $(0,2\pi)\times(-\pi/2,\pi/2)$ with periodicity ...
- 1,077
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144
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What's the best way to handle a quadratic constraint
What is the best way to handle a constraint of the type $ax_1^2+x_2^2+...+x_n^2=c$ in a gradient descent algorithm?
I would like to solve an optimization problem of the type:
$$ \min J(x_1,..,x_n)$$ ...
- 1,077
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0
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404
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How much better a bounded BFGS is compared to augmented Lagrangian method with BFGS?
I mean, in handling boxed constraints?
In terms of stability, and more importantly, the numerical performance?
I have already written some well-optimized and well-tested C/CUDA/C++ codes for ...
- 341
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751
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BFGS methods for constrained elasticity problems
My dear community,
I am wondering why BFGS methods are not so widely used for simulating mechanical problems which heavily still relies on inverting the hessian matrix. I am essentially interested ...
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