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Questions tagged [constraints]

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

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“Convert” quadratic constraint to quadratic objective

I have a large sparse quadratic optimization problem with a single quadratic constraint: $$\begin{array}{ll} \text{maximize} & c'x\\ \text{subject to} & l \leq Ax \leq u\\ & l_q \leq x'Qx ...
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Constrained Optimization using Lagrange Multipliers: Hard Problem

My problem is to maximize a the function below: $$ \frac{\left\langle\overrightarrow{\omega}_{m}^{^{^\ddagger}}\textbf{T}_{12} \overrightarrow{\omega}_{s}\right\rangle }{\sqrt{\left\langle \...
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Finding a constraint on one variable of a multivariable function to constrain the entire function

I have a function. Now I want to let my variables only take values between 0, and 1. The problem is as follows. For what values of Y, is L(x,y) < 0. That is, without putting a further constraint ...
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Can the simplex method be used for general monotonically increasing objective functions?

The simplex method relies on the fact that if you have a linear objective function and linear constraints, the optima must lie on one of the vertices of the simplex formed by the constraints. Let's ...
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23 views

A convex objective function will be convex regardless of constraints?

Let's say I have a convex objective function. The boilerplate example is $z=x^2+y^2$. Now, I also have some constraint, $f(x,y)=0$. Is it true that the constrained optimization problem must be convex ...
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Minimum value of $(x^2+y^2)^2$

if $x,y$ are real number such that $x^2+2xy-y^2=6$ Then find minimum value of $(x^2+y^2)^2$ what i try : $x^2+2xy+y^2-2y^2=6$ or $(x+y)^2-\bigg(\sqrt{2} y\bigg)^2=6$ put $\displaystyle (x+y)=\sqrt{6}...
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Proving the learnability of XOR function by a particular neural network

Let's say I have the following neural network and the constraints: The architecture is fixed (see the network in this image, I'm not allowed to post images due to low rep) (note that there are no ...
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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....
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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 $r \geq b_i \geq a_i \geq 0$...
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Conditional constraint activated by binary variables

I have the following situation in a Mixed Integer Program: $x_1, \dots, x_n$ are binary variables, and $y, z$ are continuous. If $k$ or less variables $x_i$ are set to $1$, then I need to have $y \leq ...
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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, \...
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Research articles on Multi-Objective Non-Linear Programming (MONLP)

I'm looking for papers dealing with multi-objective non-linear programming which could help me implement an algorithm to solve my problem. My problem is : Maximize $f(x) = c \cdot x$, while ...
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How to solve for two variables $10x + 20y \geq 10203$

I have a few equations of the form $$10x + 20y \geq 10203$$ Basically just two variables set equal to a value. Each equation is unrelated. So basically the goal is to find an x and y that is equal ...
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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\...
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26 views

Lagrangian dual and matrix constraints

I am starting to work with matrix calculus and I am trying to write the correct dual for the following minimization problem: $$\begin{equation*} \begin{aligned} & \underset{X}{\min} & & ...
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Find the unit vector within a subspace with the minimum norm projection onto another subspace

Let $W$ and $V$ be subspaces of $\mathbb{R}^n$ with dimensions $m$ and $p$ respectively. I want to find the unit vector in $W$ whose projection onto $V$ has the minimum Euclidean norm. From geometric ...
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How to implement the Kutta condition for potential flow and FEM?

I have a potential flow FEM solver (basic Laplace equation) which works well and is validated for potential flow around cylinder and symmetrical airfoils (such as NACA0012). However, when the angle of ...
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How to solve this set of integral equations? [closed]

Given some known functions $f(x)$ and $g(x)$, is it possible to find all solutions $\psi(x)$ that satisfy the following constraints? $$1=\int_{-\infty}^\infty dx\,f(x)\psi(x)$$ $$1=\int_{-\infty}^\...
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Forming a constrained optimization from a given problem?

A workshop produces a product out of three infinitely divisible ingredients X, Y , and Z. The ingredients cost p, q, and r pounds per kg, respectively, with p, q, r > 0 all different, and the factory ...
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how to model if else statement in mixed integer program

I am trying to model a if-then condition for a MIP. The MIP looks like Maximize $\sum\limits_i H_i - C$ s.t. $\sum\limits_j x_{ij} \le D_i$ and $\sum\limits_i x_{ij} \le S_i$, where $H_i = 1$ if $\...
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Big M Equality Constraints Question

I am a newcomer to mixed integer linear programming, and I am having some trouble using the Big M method to linearize some constraints, and was wondering if I implementing it incorrectly. Here is ...
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Equivalence of two MLE estimators with a constraint

Consider the following objective function $S\left( \theta ,w|\Lambda \right) .$ The value of $S\left( \theta ,w|\Lambda \right) $ depends on the nuisance parameters $\Lambda .$ However, the first ...
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Prove Unique Lagrange Multipliers Equality Constraint

I am working through some old test papers in preparation for exams an am trying to scout out potential sneaky questions that might be asked. I've stumbled across this one. Would you please verify or ...
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knapsack with multiple constraints and some negative weights

I'm trying to solve an integer linear programming problem of the following form $max$ $\sum_{i=1}^n v_i \cdot x_i$ $s.t. \sum_{i=1}^n w_{i1} \cdot x_i \leq 0$ and $\sum_{i=1}^n w_{i2} \cdot x_i \...
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1answer
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find extrema of a multivariable function Lagrange

I need to evaluate the extrema of this function : $f(x,y,z)=\frac{1}{x^2+y^2+z^2}$ restricted on: $R=\{x^2-y^2-z^2+16<=0\}$ \ $\{0,0,0\}$ boundary : $\theta R=\{x^2-y^2-z^2+16=0\}$ \ $\{0,0,0\}$ ...
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39 views

find extrema of a $f(x,y,z)$ function using Lagrange multiplier

The function is : $f(x,y,z)=e^y(x^2+z^2)$ restricted on $R=\{x^2-3y^2+z^2+9=0,x^2+y^2+z^2\le 16\}$ $$ \left\{ \begin{aligned} 2xe^y=\lambda 2x+\mu 2x \\ e^y(x^2+z^2)=-\lambda 6y+\mu 2y\\ 2ze^y=\...
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find extrema min,max on a multivariable function

I have to evaluate min,max inside and on the boundary of a domain: $D=\{xy-1\le 0,|y-x|\le1\}$ $f(x,y)=(y-x)e^{xy}$ So That's a ($xy-1$) hyperbola and two lines. I proceeded like so: for $y=1+x $ ...
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How to use non-binary variable in a conditional statement MILP?

I have a conditional statement I want to implement in a MILP. $A$ is a non-binary variable that has known upper and lower bounds. $B$ is a known parameter. And $C$ is a binary variable. How do I ...
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48 views

Solving dynamic optimization with non-binding inequality constraint

I want to solve a problem similar to the following discrete and finite time horizon dynamic optimization problem : \begin{equation} \begin{split} &\max_{\{d_t\}} \sum_{t=0}^{T} - \left [ f(s_t) + ...
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Adding the sum of absolute values must be equal to 1 as a Constr with a linear solver

I have a big problem with Constr. i was: $$\min \sum \sigma_{x_i}$$ $$s.t \sum x_i = 1, where \space \space0<= x_i<=1$$ and now I need that problem to be: $$\min \sum \sigma_{x_i}$$ $$s.t \...
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Constrained Optimization Insights

I have been experimenting with the following problem paraphrased from Khan Academy: A manufacturer's revenue is $100h^{2/3}s^{1/3}$, where $h$ is the number of hours of labor hired, and $s$ is the ...
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1answer
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How do I define a constraint in such a way that values must be in sequential columns?

I am trying to formulate a linear program for a time scheduling problem. My variable is simple, $x_{ij}$, which is equal to 1 when job $i$ is done at hour $j$. Now as a part of this schedule (10 job ...
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76 views

Constrained Optimization Geometry Confusion

In a constrained optimization problem, let's consider the example $$\begin{cases}f(x,\ y) = yx^2\ \Tiny(function\ to\ be\ maximized) \\ g(x,\ y) = x^2 + y^2 = 1\ \Tiny(constraint)\end{cases}$$ why ...
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Count the number of unique elements in a vector by linear constraints (ILP)

Let $\mathbf{x}\in \{0,1\}^n$, be the objective variables of an ILP. Further, let $\mathbf{a} \in \mathbb{N}_{\geq 0}^n$ be a given random vector and $\mathbf{w} = \mathbf{x} \odot \mathbf{a}$ where ...
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Show that $C=\{x \in \mathbb{R}^4 \mid x^TAx \geq 1\}$ is not empty?

Consider the set $C=\{x \in \mathbb{R}^4 \mid x^TAx \geq 1\}$ where $A \in \mathbb{R}^{4 \times 4}$ is a symmetric matrix with two distinct positive eigenvalues and other eigenvalues of $A$ are ...
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System of ODEs with integral constrains

Can someone point me in a direction to solve this kind of integral constrained system of ODEs. As far as I know, there are no analytic methods that can solve this. So I will resort to numerical ...
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1answer
70 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 ...
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24 views

McCormick envelope of two variables which are also defined in terms of an envelope

I have a equation which is defined as $\langle\langle x_ix_j\rangle^M\langle \cos(\theta)\rangle^C\rangle^M$ where $M$ is the McCormick envelope of product of variables $x_ix_j$ and $C$ is defined as ...
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1answer
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Optimal transport with relaxed constraint on terminal distribution

I have read the topic on relaxing constraint on relaxing marginal constraints Optimal transport with relaxed constraint on marginals, where the constraint is expressed as the difference of initial and ...
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1answer
37 views

The shape of a feasible region with equality and inequality constraints

I was wondering if anyone can help me with this (probably basic) question. I want to know how the following feasible region looks like if we have thousands of variables. The constraints are linear. ...
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1answer
38 views

Solving for integrand from integrated quantities.

Given equations of the form: $A(r) = \int_{t_{1}}^{t_{2}}F(r,t)dt$ $B(t) = \int_a^b F(r,t)r^2dr$ where $A(r)$, $B(t)$, and all of the limits on the integrals are known, is there enough information ...
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176 views

Norm constrained least square minimization with an additional single linear equality constraint: A quadratically constrained quadratic program (QCQP)

Given is the following QCQP problem: \begin{align} &&\min_{\mathbf{x} \in \mathbb{C}^n} &\|A \mathbf{x} - \mathbf{b}\|^2,\tag{1}\label{1}\\ \text{ subject to:}&&&\\ &&\|...
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1answer
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Number of independent components of a vector satisfying a differential constraint?

Edited question Consider a vector field $\vec{A}(\vec{x})$ such that in one case $\nabla\cdot\vec{A}=0$. It looks like that this condition gives rise to a differential equation constraint $$\...
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1answer
46 views

Least Squares Solution of Minimal Norm when $A^{*}b = 0$

Suppose, given a matrix $\textbf{A} \in \mathbb{C}^{m \times n}$ and a vector $\textbf{b} \in \mathbb{C}^{n}$, I want to find the minimal norm solution of $$\min_{\textbf{x}}\|\textbf{A}\textbf{x} - \...
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3answers
111 views

Binary variables in time series: integer linear programming

I'm working on a problem and I can't seem to find an easy solution to it. It's about an optimization problem, concerning a time series. I have a binary variable $\alpha_t$ for $t \in [0, 24[$. I ...
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Conditions for two B-Splines to represent the same curve

What will be the weakest constraint for two B-Spline curves $S_1(t)$ and $S_2(u)$ to represent the same curve in space? Assume that their orders are $k_1$ and $k_2$ respectively, knot vectors are ($...
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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 ...
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18 views

$L_{2,0}$ constraint in the optimization problem

I am trying to solve a minimization problem where my constraint is $$||W||_{2,0} = 1$$where $W \in R^{k \times k}$. The constraint is used in such a way that only one column of $W$ matrix is non-zero. ...
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1answer
24 views

Minimise $C(x,y)=11x+3y$ subject to the constraints.

Minimise $C(x,y)=11x+3y$ subject to the constraints $ g(x,y)=-3x^2-3y^2+10xy $ and $x\geq 0, y\geq 0$. I started solving using this Lagrange multiplier, but the constraint set is not compact, right? ...
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Sampling from joint discrete distribution

I have a set of items $a_1, a_2, \dots, a_n$. My aim is to generate from this set of items, a list of item tuples $\{(a_i, a_j), \dots\}$ such that $a_i\ne a_j$. The constraints are as follows. The ...