Questions tagged [constraints]

In mathematics, a constraint is a condition of an optimization problem that the solution must satisfy.

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6
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1answer
544 views

Integer linear programming constraint for maximum number of consecutive ones in a binary sequence

Consider an integer programming problem with binary decision variables $x_1,\ldots,x_n \in \{0,1\}$. Im trying to model the constraint that enforces the maximum number of consecutive ones in ...
5
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0answers
202 views

Vector Laplace equation with constraint

I want to solve Laplace equation for a vector $\boldsymbol v=(v_x,v_y)$: $$\nabla^2 \boldsymbol{v}=0$$ but under the constraint that $$(1+v_x)^2+v_y^2=1$$ which becomes $v_y = -(2v_x+v_x^2)^{1/2}$. ...
4
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0answers
49 views

Momentum constraints for a Singular Lagrangian

Note I've explicitly indicated it at points in this question, but unless stated otherwise $i,j,k \in \{1, \ldots, n\}$, $a,b,c \in \{1, \ldots, R_W\}$, and $\alpha, \beta, \gamma \in \{R_W + 1, \...
4
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0answers
61 views

projection from a point to a constrained hyperplane

I am trying to find the closest point on the following constrained hyperplane to a general point $\vec x$ : $$ \vec \omega \!\cdot\! \vec 1 = 1 \ \ s.t \ \ \alpha_i \le\omega_i\leq\beta_i $$ $$ 0\...
4
votes
2answers
114 views

How to take the partial derivative of $f(x,y) = x\ln(x) + y\ln(y), x + y = 1$?

Let $f(x,y) = x\ln(x) + y\ln(y)$ be defined on space $S = \{(x,y) \in \mathbb{R}^2| x> 0, y > 0, x + y = 1\}$. My question is, how do I take the partial derivative for this function, given ...
4
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0answers
45 views

How to apply Newton's method to solve for a root of a function that is not everywhere defined?

If I want to solve for a root of a nonlinear function $f$, I would naturally consider using Newton's method, starting from an initial guess $x_0$: $$ x_{k+1} = x_{k} - \frac{f(x_k)}{f'(x_k)} $$ Now, ...
4
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0answers
166 views

Calculus/Optimization - Implicit fuction theorem with equality constraints

I have the following constrained maximization problem, written as a Lagrangian: $$ L(x,y,\lambda) = f(x,y) - \lambda(g(x,y)) $$ I can derive a set of implicit equations that characterize the solution, ...
4
votes
1answer
239 views

Asymptotics of Gaussian integral over the unit sphere

I would like to evaluate the integral asymptotically over the unit sphere surface $$ Z =\int e^{a \cos^2 \theta + b \sin^2\theta\cos2\phi + c\cos\theta} d\Omega = \int\limits_{0}^{\pi}\int\limits_{0}...
3
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0answers
91 views

Riemannian Manifold for the Partial Doubly Stochastic Matrices

Excuses if my formulation is non-rigorous. I am not a mathematician by training. I have a constrained optimization problem where each of my matrix valued parameters lives inside the Birkhoff Polytope....
3
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0answers
31 views

Computing hyper area of a contrained simplex

Let $D_n [r , (a_1, b_1 ) , \ldots , (a_n, b_n) ] = \{ (x_1 , \ldots , x_n ) \in \mathbb R^n \mid \sum_i x_i = r \mbox{ and } b_i \geq x_i \geq a_i \, \forall i \}$, where $r \geq b_i \geq a_i \geq 0$...
3
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1answer
78 views

constrained rank approximation

I'm trying to solve a problem similar to this problem. Instead of requiring the diagonals to be 0, I'd like to require columns of the low rank approximation to decrease in value while going down the ...
3
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0answers
66 views

Mathematics of Putting (finding function for ball rolling on surface under gravity)

I was thinking up a problem today, and am unsure how to solve it. Let's say we have a surface modeled by $g(x,y,z) $, where $z $ is the height from the lowest point, which we will set as $z = 0$. ...
3
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0answers
627 views

Constraints on a matrix

I have a matrix equation of the form $A \times B = C$. $A$ is an $n \times m$ matrix. $B$ is an $m \times 1$ matrix. $C$ is obviously $n \times 1$. Here are the constraints: All values are positive ...
3
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0answers
537 views

How to reduce the number of (overlapping) constraints in a linear program?

I am trying to solve a linear program with more than 7 million constraints which could not be solved on my computer (In total around 5000 variables). In the constraints there is a overlap between them....
3
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1answer
63 views

Constrained nonlinear optimization

I am wondering what is the easiest/best way to find the values of $x_i$ that maximize the expression $\sum_{i=1}^N a_i \ln (x_i)$ under the constraints $\sum_{i=1}^Nx_i = 1$ and $ 0\leq x_i \leq 1$ ...
3
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0answers
769 views

From constrained to unconstrained maximization problem

I have the following constrained maximization problem $$ \max_{X_1,X_2,...,X_i,...,X_N} \sum_{i=1}^{N}X_i f_i(X_1,...,X_N) \hspace{0.2 cm} \text{subject to} \sum_{i=1}^{N}X_i-B\leq 0 \text{ and } X_i\...
2
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1answer
28 views

How can I adjust the coefficients in the constraints of a Linear Programming problem with no objective function until I get a solution?

I have a system of linear equations that I need a solution for that is strictly positive. I have 4 solutions and 4 unknowns, and the solution I obtain for my current system involves negative numbers. ...
2
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0answers
44 views

Stable numerical integration of a PDE with a bounded variable

My colleagues and I are trying to numerically integrate a physics-based PDE (phase field model of diffusive phase separation) of the form $$ \frac{\partial \phi}{\partial t} = \frac{\partial}{\partial ...
2
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0answers
48 views

Cauchy problem and boundary conditions in electromagnetism

Consider a connection on a principal $U(1)$-bundle $A_\mu$ over the flat base manifold $M_4$. The action of the theory is described in terms of the curvatures of such connection coupled to some source ...
2
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0answers
70 views

Combining inequalities to not have coefficients

We have in input an inequality cons and a set of inequalities C and we want to find a way to sum them and simplify in a way that no variable has a coefficient and that the number of constraints that ...
2
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0answers
24 views

When will $l_1$-relaxation give the same answer as the $l_0$ problem?

The $l_0$-relaxation problem $$\min_x \Vert x \Vert_0 \text{ subject to Ax = b}$$ is non-convex. On the other hand, the $l_1$ problem $$\min_x \Vert x \Vert_1 \text{ subject to Ax = b}$$ is ...
2
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0answers
37 views

Apply integration type constrain condition in FEM

I'm running some simulation use FEM, in my model, I need to apply a constraint condition to the governing equation. My governing equation similar to the diffusion equation as below: $$\frac{\partial ...
2
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0answers
339 views

LASSO relationship between Lagrange multiplier and constraint and why it doesn't matter

My understanding of LASSO regression is that the regression coefficients are selected to solve the minimisation problem: $$\min_\beta \|y - X \beta\|_2^2 \ \\s.t. \|\beta\|_1 \leq t$$ In practice ...
2
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0answers
156 views

how to minimize constrained Frobenius norm equation

I’m trying to understand what is the exact process to minimize the following constrained Frobenius norm equation: $$ ||A-CW||_{F}^2 $$ s.t. $$ WW^T\ge D $$ I understand one possible way is to use the ...
2
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0answers
55 views

Network flow paths with parameterized capacity constraints

I'm trying to find a solution to a problem: I'm given a connected network $G=(V,E,c)$, where c is an edge capacity. I also have pairs of terminal nodes $s_i\in V$ and $t_i\in V$, a flow value $f_i\in ...
2
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0answers
52 views

Constraint optimization problem with an unusual constraint

I struggle with solving the following minimization problem: $F=||X-(A+B)C^T||_F^2$ s.t. $A⊙B=0$ where ⊙ is Hadamard product. My solution is as follows: First, getting Lagrangian form of $F$: $L=||...
2
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0answers
38 views

Problem with defining a constraint where a weight has to be included

I need to write up a model for a scheduling problem using linear integer programming. This goes well so far but I am stuck with one constraints that I do not know how to write up. I will try to ...
2
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0answers
352 views

Constrained least squares

Suppose that the least squares problem $$\min_x \| Ax-b \|$$ has a unique solution $x^*$. Now, when we consider the constrained version such as ${\rm min}_x$ $\parallel Ax-b\parallel$ s.t. $x_i \...
2
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1answer
726 views

Minimisation in Linear Programming

I'm somewhat stuck on an example in Linear Programming. I managed to wrap my head around maximisation for a problem with $\le$ constraints, using both graphical and simplex solutions. However, I have ...
2
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0answers
266 views

Lasso with non-linear objective

I have a non-linear objective function that I want to minimize considering some constraints in order to obtain a sparse solution (lasso type). min f($\theta$) s.t. $\sum_i|\theta_i|\leq t$ $\theta_i ...
2
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0answers
607 views

Constrained Non-Linear Least Squares Solver

I need C# code for solving constrainted non-linear least squares problems. I'm prepared to write the code myself, but I need to understand the algorithm first. Can anyone describe a constrained non-...
2
votes
1answer
186 views

How to deal with an $xy\le 1$ constraint?

I have to solve the following optimization problem: $$ \begin{align*} \min_{x,y} &\{-x-y\} \\ \text{such that} \\ y &\ge 3 \\ y &\le 30 \\ x &\ge 0 \\ xy &\le 1 \\ \end{align*} $$ ...
2
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0answers
62 views

How to deal with a convex constraint

I want to deal with a convex constraint \begin{align} F(P)=P^{H}AP_{0}+P_{0}^{H}AP-P_{0}^{H}AP_{0}\succeq 0 \end{align} where $(\cdot)^{H}$ represents Hermitian transpose, $A$ is a positive definite ...
2
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0answers
365 views

Generalized Bezout's Identity, subjected to a constraint on coefficients

I would like to know if there exists any good methods that can determine the following class of problems: Suppose there exists $n$ given positive integers $y_1, y_2, \dots, y_n$ and positive integers ...
2
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0answers
70 views

Rook Polynomials with Symmetrical Overlap (Count Permutations Restricted by Distance)

Consider the cardinality $P(n,d)$ of permutations where elements can move up to distance $d$; for example, the permutation $\binom{012}{102}$ with $d = 1$ would be valid but $\binom{012}{201}$ would ...
2
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0answers
44 views

Solution to a combinatorial constraint system

I am facing a combinatorial problem where I am interested in the minimum number of constraints of a certain type that uniquely determine a solution. I realize that my problem is highly specific (and I ...
2
votes
1answer
566 views

Inhomogeneous eigenvalue problem, the shooting method and constraints

In trying to solve a problem occurring in QM calculations I've encountered the following pickle, with which I hope you could help me. I am trying to solve an inhomogeneous eigenvalue differential ...
2
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0answers
808 views

solving euler-lagrange equation in constrained functional optimization

The problem to solve is the minimization of a functional of two functions, $F(y,z) = \int_a^b f(y,z)dx$ , subject to a constraint $g(y,z,y',z') = 0$. The augmented functional is then $L(y,z,y',z') = \...
2
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0answers
184 views

ODE with constraints

Given the ODE system $$\dot{x} = y \\ \dot{y} = \frac{1}{\alpha} (z - y)$$ where $\alpha > 0$ is a constant. How can I find a bound for $z$ depending on $x$ such that $\forall t ~x(t) \geq 0$ under ...
1
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1answer
36 views

Iteratively partitioning a set into $k$ equally sized subsets where each pair of members occurs in a subset at most $x$ times over all partitions

I would like to partition a set ${1, 2, ..., n}$ into $k$ equally sized subsets, and perform this operation $b$ times. In the end, my aim is to end up with a situation where for each given pairwise ...
1
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0answers
23 views

Optimization problem with inequality constraints

Suppose we have $\theta=(\theta_1,\ldots,\theta_n)$, with $v_i:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ being a continuous, differentiable, concave function. Now I want to solve the following ...
1
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0answers
46 views

Constrained optimization using function of function

Suppose I have the following constrained optimization problem $$max \quad f(x) \quad s.t. \quad g(x)=a$$ whose solution is denoted by $x^{*}$. I want to prove that this is the solution to the above ...
1
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0answers
59 views

Find the desired function or disprove its existence

Let $T(n,m)=\frac { n^2\cdot m\cdot f(n)}{n!}$. I need to find $f$ in terms of $n$, such that: $f$ is non decreasing function $f(n)\in\Omega(1)$ $\exists k>0.\ f(n)\in O(n^k)$ The following ...
1
vote
1answer
33 views

Changing a strict inequality to a non-strict inequality?

Is there a way to change a strict inequality (e.g. >) into a non-strict one? (e.g. greater than or equals to)? If not, how would I deal with this problem? I have been attempting this and reached the ...
1
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0answers
38 views

Why does the minimum of F correspond to the lowest eigenvalue of L?

I have been studying variational principles and I have been reading this set of notes. In section 7.1, we study the Sturm-Liouville problem, as described below. Let $p(x)$, $\sigma(x)$, $w(x)$ be ...
1
vote
1answer
37 views

why are the extreme points equivalent to basic feasible solutions in a linear programming problem?

Let extreme points be the set X={x greater or equal to 0 given Ax=b for vector x and b} and a point x is extreme if for all y,z in X, x=(1-a)y+az for a=[0,1] Basic Feasible solution x is if for A be ...
1
vote
1answer
48 views

Using Kuhn-Tucker to determine if constraint is binding

Let $f(x,y,z)=xyz+z$ be the objective function, and suppose that $0\leq6-x^2-y^2-z$, $0\leq x$, $0\leq y$, $0\leq z$ are the four constraint functions. I must determine if the constraint $0\leq 6-x^2-...
1
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0answers
30 views

How does the integral constraint for a PDE over a volume equaling zero ensure uniqueness?

I am studying partial differential equation optimization, and this is the elliptic model problem: (1) $- \nabla \space \cdot (e^u\nabla y_i)=q_i\space\space\space x\in\Omega$ (2) $\nabla y_i \space \...
1
vote
1answer
36 views

Constraint formulation that include consecutive values in an optimization problem

I am currently lost in finding a way on how to mathematically formulate a constraint within the following problem: I want to allocate some water tanks locations within a network, which size will vary ...
1
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0answers
44 views

Convexifying the biconvex constraint $xy-f(x,y)=0$ to $\max f(x,y)$

I want to solve $$\min_{x,y} \quad -f(x,y) \qquad \text{subject to} \quad xy C - f(x,y)K -\kappa = 0$$ where $x, y$ are from a compact and convex subset of $\mathbb{R}^2$ $f(x,y)$ is concave, in $...