Questions tagged [constraints]

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

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Existence of global minimum $f(x,y,z) = x + y + z$ under the constraint $x^2+xy+2y^2-z=1$

The full exercise consists of (i) finding the minimum value of $f(x,y,z) = x + y + z$ under the constraint $g(x,y,z)=x^2+xy+2y^2-z=1$, and (ii) establishing whether the function has a maximum. I have ...
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0answers
24 views

Matching position and rotation of moving target.

So I'm trying to work out how to intercept a moving rotating target. The key is that that I must match both $x$, $y$ coordinates and the rotation of the target. I'm a little confused, as this must be ...
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1answer
274 views

Second-order cone constraints

If we have a system of constraints given by, $$Ax \preceq_K b$$ where $K$ is a second-order cone, would this simply be the same as requiring that: $$\|Ax\|_2 \leq b$$ where $\|\cdot\|_2$ is the $2$...
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97 views

Convexify this optimization problem with one nonlinear (bilinear) constraint

I have the following nonlinear optimization problem: $$ \begin{aligned} & \underset{R,\theta,f,s}{\text{minimize}} && \sum_{i=1}^m L_iR_i^2 \\ & \text{subject to} &...
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1answer
131 views

MILP constraints with truth table

I have two decision variables, $SL_i$ = the amount I'm contributing to my student loan debt in month i, and $fiftySL_i$ = 1 if I contribute at least \$50 over the minimum payment (so contribute $350) ...
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2answers
83 views

Constrained optimization where the choice is a function over an interval

I would like to solve a constrained optimization problem where the choice is a function over an interval rather than a finite number of variables or a sequence. The problem is given by: $\max_{[x(i)]...
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114 views

Constrained eigenvalue problem

what is the typical way of solving an eigenvalue problem if you have additional constraints?? Let's say for example $$ \left(\begin{matrix}a && b && c\\ d && e &&f \\ ...
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4answers
302 views

Find all local maxima and minima of $x^2+y^2$ subject to the constraint $x^2+2y=6$. Does $x^2+y^2$ have a global max/min on the same constraint?

This is my method for the local max/min. Does this answer sound sensible? (Not sure how to go about checking for global max/min though...) Method $G(x,y)=x^2+2y-6$. Rewrite in terms of $y$, so $y=((-...
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1answer
69 views

Optimization - If the sum of objective functions are similar, will sum of argmax's be similar

Suppose that there are $n$ distinct objective functions $f_1,..,f_n$. And another $n$ distinct objective functions $g_1,..,g_n$. Each $f_i: X \to R$, and each $g_i: X \to R$, where $X$ is some set (...
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660 views

Convex problem with concave constraint

I have a simple question because I want to optimize an objective function which is non-linear and convex but is limited by 3 non-linear constraints, two of them are convex but the last is concave. ...
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Lagrangian Constraint Help - why is this the answer?

Q: a factory produces bodies & wheels for standard cars. Each car has to be fitted with one spare wheel. Total number of wheels produced is denoted by W. Number of car bodies is B. Profit ...
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155 views

implicit differentiation when the implicit equation is a differential equation

I want to find derivative/variation (I think variation is more correct) of solution of a differential equation with respect to one parameter of differential equation but without solving the ...
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54 views

How can I express this as a linear constraint?

Let $v \in \mathbb{R}^n$ be a known vector. Let $Q \in \mathbb{R}^{n \times n}$ and $s \in \mathbb{R}$ be decision variables of an optimization problem. Let $Q$ be a positive semidefinite symmetric ...
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1answer
39 views

Optimization problem: Choose indices to max $A$ under constraint $B$

I have the following data set: \begin{array}{|c|c|c|} \hline \text{Index}& A & B \\ \hline 0 & a_0 & b_0\\ \hline 1 & a_1 & b_1\\ \hline 2 & a_2 & b_2\\ \hline ... &...
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35 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 ...
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1answer
42 views

Find all $P_0= (x_0, y_0, z_0)$ on $z = x + y^2$ so that the angle between normal vectors at $P_0$ and $(0,1,0)$ is $\pi/4$.

Question: Find all $P_0= (x_0, y_0, z_0)$ on $z = x + y^2$ so that the angle between normal vectors at $P_0$ and $(0,1,0)$ is $\pi/4$. My attempt: Let $F(x,y,z) = z - x - y^2$.Then the condition ...
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296 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 ...
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25 views

Generating matrix where each element needs to occur on the same row as all the other elements exactly once

I need to create a matrix where each element in the matrix will be on the same row as every other element exactly once. An example with the number 1-9 can look like this: $\begin{bmatrix}2 & 4 &...
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1answer
90 views

What does a function inside brackets and a minus sign mean?

I am working in constraint optimization. And I've just come across this notation. I'm not sure what this means. The set I is the set of inequality constraints. I'm not sure what the function in ...
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1answer
649 views

Why use Bordered Hessian than “simple” Hessian as second derivative test?

Why use Bordered Hessian than "simple" Hessian as second derivative test in a multi constrained optimization problem? The critical points are found from the Lagrangian so they follow the constraints. ...
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1answer
119 views

Convert NL equality constraint involving minimum to linear inequality constraint?

Is it possible to convert an equality constraint involving the minimum, to a linear inequality constraint? Suppose I have an optimization problem which involves the variables $x_1,\,x_2,\,x_3$, with ...
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1answer
21 views

primal constrain

I'm new to the dual problem. As you see, here gives one primal problem. I'm confusing about the constraint $3\leq x_3 \leq 4$, in the textbook I can not find the similar one. Could anybody teach me ...
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2answers
134 views

Lagrange Multipliers: “What is a Critical Point?”

So we have function $f(x,y) = e^{xy}$ this has the constraint $x^3+y^3=16$ By use of Lagrange Multipliers or the Lagranian we find there is only one critical point at (2,2). What confuses me alot is ...
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169 views

What is the formula for a quadratic curve with defined crossing and maxima points?

This is a problem that I've been wrestling with on and off in the process of creating quadratic splines for a game I'm building. Given crossing points $x_0$ and $x_1$, and a maximum point $m$, how ...
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30 views

Summation expansion

I have the following sum: $$F=\sum_{i=1}^N p_i g_i .$$ Where $g_i$ is random variable and $p_i$ is function of $g_i$ such that, if $g_i<0.01 , p_i=0$. On the other hand, I have (CONSTRAINT): $$...
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1answer
53 views

I need to prove the monotony and the constraint of the sequence $(a_n)$, and also find its boundary if $a_1= 3/2$; $a_{n + 1}^2 = 3a_n - 2, n ≥ 1$

I need to prove the monotony and the constraint of the sequence $a_n$, and also find its limit if $a_1 = 3/2$; $a_{n + 1}^2 = 3a_n - 2, n ≥ 1$. Should I use the method of mathematical induction here?
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1answer
31 views

I don't understand the constraints for this scheduling problem

In a flexible job-shop scheduling problem, we are trying to minimize the makespan. We have $n$ jobs that need to run on $m$ machines. Each job $i$ consists of $n_{i}$ operations ($O_{i1},O_{i2},…,O_{...
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1answer
287 views

How to convert linear inequality constraints into box constraints

I am trying to solve an MPC type problem in which I have equality constraints of type $x(k+1) = Ax(k) + Bu(k)$ which are because of system dynamics and I have inequality constraints of type $h_{min}...
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340 views

How to find Nash equilibria through KKT conditions (convex optimization)?

I have a game defined by the following utility functions (example with two agents): $v_1 = u(w_1 - x_1 + \alpha(x_1 + x_2)) + \beta_{12} u(w_2 - x_2 + \alpha(x_1 + x_2))$ $v_2 = u(w_2 - x_2 + \alpha(...
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0answers
36 views

How to convert a sum and product constraint into SMT-lib2

I'm wondering what the best way to convert a sum a le $$\sum_{v=exprLB}^{exprUB} 2 + exprContent(v) = 12.34$$ or similarly a product $$\prod_{v=exprLB}^{exprUB} exprContent(v) = 1234$$ into an SMT-...
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0answers
148 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 ...
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57 views

Express union of constraints with inequality system

How can we express union of two or more constraints, for example $x\ge 0 \vee y\ge 0$, in an inequality system as below: $f_1(x,y)\ge 0 \\f_2(x,y)\ge 0 \\f_3(x,y)\ge 0 \\ \qquad\vdots$ where $f_i\in ...
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1answer
694 views

Visualizing the constraint matrix in an integer linear program

Suppose we have an integer linear program of the form: $\begin{equation*} \begin{aligned} & \text{minimize} & & \sum\limits_{i=1}^n \sum\limits_{j=1}^n c_{...
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1answer
194 views

Optimization Software for mathematical models with (arg) min/max in constraints

With context of a college student timetable and course selection, I’m formulating a function that counts holes (the empty blocks of hour when there is no class) limited by the first course of the day ...
2
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1answer
37 views

What distributions can I simulate with this stochastic process?

For fun, I am trying to use a two-step stochastic process to simulate an existing distribution $\text{P}(x_i) = p_i > 0$ on some discrete set $x_1, ..., x_n$. The process is as follows. Pick an ...
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1answer
181 views

Minimize convex function with concave equality contraint

Let (1) $f:R^k \rightarrow R$ be a convex and differentiable function and (2) $g:R^K \rightarrow R$ be a concave and differentiable function. Consider the minimization problem $\{\min _x \ \ f(x) \ \ ...
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Expressing “OR” operator as linear constraint for optimization [closed]

I am formulating optimization problem which has to satisfy ( $ 20°\leq\theta_{1}\leq160°$, $ 200°\leq\theta_{2}\leq340°$, $ 200°\leq\theta_{3}\leq340°$ ) OR ( $ 20°\leq\theta_{4}\leq160°$, $ 200°\leq\...
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70 views

How do I formulate a sum constraint in quadratic programming?

I am attempting to solve a quadratic programming problem of the form: $$\mathrm{min} \ \frac{1}{2}\alpha^T G \alpha$$ $$ \mathrm{s.t.} \ \sum_{i=1}^{n} \alpha_i y_i = 0$$ $$0 \leq \alpha_i \leq C$$ ...
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1answer
31 views

Spheres in higher dimensions …

Original Problem: For a given $y$ and $n$, I want to find all $x_i$'s that satisfy \begin{equation} \frac{x_1^2}{\sum_{i=1}^{n}{x_i^2}}=y \tag{1} \end{equation} while satisfying following ...
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1answer
308 views

Constrained optimization : Contour lines and Lagrange's multiplier

Basically the core of Lagrange's multiplier says that the solution to a constrained optimization occurs when the contour line of the function being maximized/minimized is tangential to the constraint ...
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1answer
72 views

Why not differentiate the Lagrangian w.r.t a lagrange multiplier?

I've heard from a reuptable source that it is problematic to differentiate the Lagrangian w.r.t the lagrange multiplier. I know that doing so is rather a waste of time since it just goves you back ...
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2answers
54 views

2 constraint optimization (Lagrange multipliers)

Determine the critical points of $x^3 + y^3 +z^3$, such that $x^2 + y^2 +z^2 = 1$ and $x + y+ z = 0$ by hand. Attempt at a solution: I seem to figure out $-1$ as the multiplier for $x+y+z=0$ but can'...
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1answer
65 views

Multivariable Calculus finding absolute extremas with constraints?

PART ONE: I know how to find extremas by just using the gradient of a function $\nabla f(x,y)=0$ But I have been given a function alongside a constraint equation. And I am immediately thinking to ...
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2answers
268 views

KKT conditions for L2 norm

I have the following minimization problem \begin{gather*} \text{minimize} \quad ||w|| \quad \quad w\in\mathbb{R}^2 \\ \text{subject to} \quad w_1+w_2+1\le0 \end{gather*} I would like to solve it ...
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1answer
290 views

Integer programming : how to express that one linear constraint implies another?

I have formulated a linearly-constrained integer optimization problem. For now, I have been solving it by using an exhaustive search approach over the integer variables. However, I would now like to ...
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1answer
60 views

Conditions to make conic section positve in the region $0 \leq x+y \leq 1$

Suppose $0 \leq x+y \leq 1$ and let $a,b,c \in [0,1]$ be fixed. I've been stuck with this inequality: $$ -2\,{b}^{2}{x}^{2}+ \left( -4\,{b}^{2}+4 \right) xy+ \left( -4\,ab-2\, c-2 \right) x-2\,{b}^{2}...
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4answers
49 views

Constrained Optimisation Problem: Confusion with Algebra/Contradiction

I completed the following constrained optimisation problem: Maximum and minimum values of $f(x,y) = x^2 + 2xy + y^2$ on the ellipse $g(x,y) = 2x^2 + y^2 - xy - 4x = 0$. $\nabla f = \lambda \nabla ...
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1answer
51 views

Behaviour of the solutions to a power-equality

Suppose $$ a^k(1-a)^{n-k}=b^m(1-b)^{n-m},$$ where $0<a,b<\frac{1}{2}$ are real numbers, $n,k,m$ are positive integers, $0<k<m<n$. How to prove that $a<b$? I am feeling this should ...
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1answer
306 views

JavaScript QP solver doens't give correct solution.

I'm solving a constrained optimization problem, where I have to find the vector $\mathbf{x}$ of dimensionality N x 1. The input vectors are $\mathbf{a}$ and $\...
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1answer
101 views

Least squares minimization subject to the constraint $\sum_j|u_j| \leq 1$

I would like to learn about methods for minimizing the cost $L(u) = (f - Au)^{tr}M(f-Au)$, where $f \in R^m$ is a known vector, $u \in R^n$ is the argument of the minimization and $A \in R^{n\times m}$...