Episode #125 of the Stack Overflow podcast is here. We talk Tilde Club and mechanical keyboards. Listen now

Questions tagged [constraints]

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

Filter by
Sorted by
Tagged with
2
votes
0answers
35 views

Linearize if-then constraints

For continuous variables $x$ and $y$, the constraints are: ...
0
votes
0answers
8 views

Adding inequaltiy contrains in ODE's using bvp4c in Matlab

Consider following code in Matlab: ...
0
votes
2answers
34 views

Box constrained maximization using Kuhn tucker

I have to maximize the following function - Max A $C_1^{-m}/{-m}$ + (1-A) $C_2^{-m}/{-m}$ Subject to, $C_1$ ≤ 5(1-x) + x $C_2$ ≤ 3(1-x) + 7x 1≤x≤10 I wrote it as: L(x) = f(x) - $λ_1$($C_1$ - 5(...
0
votes
1answer
56 views

How can I solve this linear optimization problem?

I've come across a question which I was not able to solve I would appreciate if someone could help me out here. Q) Given the constraints, $$x \ge 0$$ $$y \ge 0$$ $$x + y \le1$$ which of the ...
0
votes
0answers
24 views

Optimal control problem with dynamic constraints having state derivatives as a function of the control derivative.

I have a question with regard to optimal control theory and specifically to an optimal control problem that has dynamic equations/constraints not in the usual format of $\dot{x} = f\left( x, u \right)$...
2
votes
2answers
89 views

Solving constrained minimisation problem using unconstrained optimization of the generalized Lagrangian

My textbook, Deep Learning by Goodfellow, Bengio, and Courville, says the following in a section on constrained optimization: The Karush-Kuhn-Tucker (KKT) approach provides a very general solution ...
1
vote
1answer
18 views

Reformulating constraint containing equivalence

I want to reformulate the constraint: $$[x = a] \Leftrightarrow [d = 1]$$ where $d$ is a binary variable $$ d \in \mathbb{Z}, \quad d \geq 0, \quad d \leq 1$$ and $x$ is bounded $$m \leq x \leq M$...
0
votes
1answer
31 views

Maximising a quadratic function under a constraint

Let $$f:[0,1]^n\rightarrow\mathbb{R},~~(x_1,\ldots,x_n)\mapsto \sum_{i=1}^{n} x_i(1-x_i).$$ Now fix $\overline{x}\in[0,1]$. I would like to show that under the constraint $$\overline{x} = \frac{1}{n}\...
1
vote
2answers
29 views

How does $Ax = b$ define a feasible region of half spaces?

When a linear program is formulated like this: $\begin{align} \text{minimise}\quad &c^Tx\\ \text{subject to}\quad &Ax \ge b \end{align}$ With $c\in \mathbb{R}^{|x|}$, $A \in \mathbb{R}^{n \...
0
votes
0answers
8 views

discretize semi infinite and identify multiple violated constraints using a random search

I have the problem of semi-infinite optimization. max{b*y : g(y,ω) ≤ 0, ω ∈ Ω} Someone can explain this to me: at each iteration an oracle is used to discretize Ω and identify multiple ...
1
vote
1answer
38 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 ...
0
votes
0answers
15 views

Different type of constraint in Lagrange multiplier method

I am studying a probability distribution function of velocity of a relativistic particle ($f(v)$). My aim is to obtain it maximizing the Shannon entropy function with some constraints. I was wondering ...
0
votes
1answer
30 views

how to proof (not necessarily in the official way) min with constraint

I know it's kind of question that the common answer is easy to see... still looking for some more official way to formulate it. $$ min_{y,x} y \ \ s.t. \ y \ge f(x) = min_x f(x) \ | x,y\in \...
1
vote
1answer
30 views

Vanishing of a function with a shifted argument?

Given a function $f(x)$ with $x\in \mathbb{R}$, does the condition $$f(x+1)=0\text{ for all }x$$ necessarily imply $f(x)\equiv 0$? I am asking, since e.g. in an instance where $f(x)$ is defined ...
1
vote
1answer
17 views

Constrained optimization: Imposing norm constrain on input to find small solutions

My textbook, Deep Learning by Goodfellow, Bengio, and Courville, says the following in a section on constrained optimization: Sometimes we wish not only to maximize or minimize a function $f(\...
0
votes
2answers
34 views

optimizing the function $(x^2+y^2)e^{-xy}$, is there a global max and/or global min?

I want to find global max and/or min of the function $$h(x,y)=(x^2+y^2)e^{-xy}$$ on the domain $\frac{x}{2}\geq y\geq2x$, or prove that it does not exist. I see by looking at the function that it is ...
0
votes
0answers
21 views

constrained optimization ; how to incorporate a constraint

I am solving an equation $log_2(1+\frac{xh_d p_d}{s})-\gamma x h_{dc} p_d$ s.t $xp_d h_{dc}<Q$ my $x$ comes out to be $\frac{1}{\gamma h_{dc} p_d}-\frac{s}{h_d p_d}$ but I dont know how to ...
0
votes
0answers
12 views

optimization problem with constraints

I solved a maximization problem by derivating the utility function $F_1$ w.r.t to $\alpha$ and setting it to zero. I got $\alpha=ab$ Now my utility function $F_1 $ is subject to a constraint $\alpha ...
0
votes
1answer
24 views

Variation of Parameters Problem Using Nonstandard Constraint

I am trying to solve $y'' + y = \tan(t) + \sin(t) + 1$ through variation of parameters using an arbitrarily chosen constraint. The homogeneous solution is $y_h = C\cos(t) + D\sin(t)$, so the ...
-1
votes
2answers
49 views

Three equal weights A, B and C of mass 2kg each are hanging on an ideal string passing over an ideal pulley. [closed]

The tension in the string between B and C is? Let’s consider B and C as one unit of mass 4kg. The tension on both sides of the string will be T. Also assume that the the acceleration is upwards for A ...
1
vote
0answers
24 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
vote
1answer
13 views

Constrained Optimization / derivative on curves equal zero / convergence

Consider the minimization Problem $$\min_{v\in\mathbb{R}^d\ :\ ||\boldsymbol{v}||=1} f(v)$$ where $f$ is smooth. Let's define the sequence $\boldsymbol{v}_n$ by $$\alpha_n=\operatorname{argmin}_{\...
1
vote
1answer
40 views

Integer programming: Constraint on binary variable based on parameter value

I have a set of employees $E$ that I want to allocate to a set of rooms $R$, with indices $e$ and $r$ respectively. A binary variable $x_{er}$ is 1 if an employee $e$ is allocated to a room $r$, 0 ...
0
votes
0answers
10 views

If i use some method to decrease the self interference,how should i rewrite the constraint?

In the MIMO full duplex energy harvest optimization,we can find the similar constraint like this: $SINR > \gamma $,this $ \gamma $ is some of SINR value,it means that my SINR want to be higher ...
0
votes
0answers
36 views

Reformulate as a convex constraint

I have a constraint of the following form $$\Lambda \left(\Lambda^\top\text{diag}(\boldsymbol{x})\Lambda\right)^{-1} \Lambda^\top = \theta I_n$$ where $\Lambda$ is an $n\times k$ matrix, $\...
0
votes
0answers
49 views

Convergence Rate of Projected Gradient Descent with Simplex Constraints

I'm trying to study the convergence rate, which is defined as $$ \lim_{k \to \infty} \frac{f(x_{k+1}) - f(x_*)}{(f(x_k) - f(x_*))^p} = R$$ (where $x_k$ is the $k$-th iterate while $x_*$ is the ...
0
votes
0answers
63 views

Find the maximum value of $f = x^2+ 2y^2$ subject to constraints $y -x^2 + 1 = 0$

I need to find the maximum value of the $$f(x,y) = x^2 + 2y^2$$ subject to the constraint $$y-x^2+1=0$$ Now I know this problem can be solved via lagrange multipliers and I have got the maximum value ...
0
votes
0answers
16 views

Uniform sampling under constraints

I have a problem which is a bit weird. At some point in my program I have to sample some kind of mapping under constraints, uniformly at random. Let's say I have a certain number of unique objects, $...
1
vote
1answer
52 views

Problem with finding the expected value of a trajectory under a constraint

I have a problem with trajectories $x(t)$ where $x(0) = x(T) = 0$ and $x > 0$ for all $t \in [0, T]$. I know the joint probability $P(x, t)$ and can find the expected $\left\langle x(t) \right\...
1
vote
2answers
33 views

How to name types of constraints?

I am writing a paper for school (HS level) and I defined different types of constraints for pragmatic reasons. The descriptions and examples are here: I would like to refer to them by something ...
1
vote
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
vote
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
34 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 ...
0
votes
2answers
68 views

Maximum Entropy Distribution with Constraint

I want to find the solution for the maximum entropy distribution with a cost constraint. The specific problem setup is as follows: Let $\bf{x}$ be a probability distribution. Let $\bf{c}$ be the cost ...
2
votes
1answer
29 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. ...
0
votes
0answers
21 views

Linearized feasible directions, feasible sets and tangent cones in constrained optimization problems

I would just like to verify some thoughts I have regarding these three aspects of the theory of constrained optimization. Is it true that the linearized feasible directions, can sometimes be ...
0
votes
0answers
34 views

Minimizing a Lebesgue integral subject to a single equality constraint

Let $(E,\mathcal E,\lambda)$ be a measure space, $f:E\to[0,\infty)$ be $\mathcal E$-measurable, $\mu:=f\lambda$ and $$E(p):=\int_{\{\:p\:>\:0\:\}}\frac1p\:{\rm d}\mu\in[0,\infty]$$ for $\mathcal E$-...
1
vote
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 ...
0
votes
0answers
25 views

What are integral constraints in the calculus of variations?

What do people mean, when they speak of integral constraints in the calculus of variations? How do integral constraints relate to the Euler-Lagrange equations? An example would be fantastic!
0
votes
0answers
8 views

Differentiable predicate to check if a set of points is inside a simple closed curve

In $\mathbb{R}^2$ I've a simple closed curve $\mathcal{C}$, and a set of 2D points $(x_1,y_1),\ldots, (x_n,y_n)$ (you can assume no duplicates, although an extension would be nice). I'm designing an ...
0
votes
0answers
35 views

How can we show the following metric regularity property (in the context of constrained optimization)

Let $X,Y$ be Banach spaces, $g\in C^1(X,Y)$ and $\overline x\in X$ such that ${\rm D}g(\overline x)$ is surjective. In Lemma 2.32 of this lecture notes, the following claim is made: There exists $c_g&...
2
votes
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 ...
0
votes
0answers
20 views

Best linear unbiased estimator of the mean with constaint

Let $X_1$, $X_2$, ..., $X_n$ be independent random variables with expectaions $\mu \in \mathbb{R}$ and known standard deviations $0 < \sigma_i < +\infty$. We consider the following estimator: $$...
1
vote
0answers
37 views

For positive, real $a$, $b$, $c$ with $a+b+c=1$, show $\sum_{cyc}\frac{a^3}{a^2+b^2}\geq \frac12$ [duplicate]

I have some problem solving the following inequality: Let's assume that $a$, $b$ and $c$ are all real, positive numbers, and $a+b+c=1$. Prove the following: $$ \frac{a^3}{a^2+b^2}+\frac{b^3}{b^...
1
vote
1answer
40 views

optimize $x^2+y$ on the constraint that $x^2-y^3=0$

Problem: Find maximum and minimum value for the function $f(x,y)=x^2+y$ on the constraint $x^2-y^3=0$. My Solution: I have started solving this by using the Lagrange method to find points where the ...
1
vote
3answers
60 views

Discrete constrained optimization problem

What would be the best way to solve -- either analytically or algorithmically (in this case preferably using Python) -- a discrete constrained optimization problem of the form $$ \vec x^\star = \arg\...
0
votes
0answers
57 views

Lagrange Multiplier with slack variable to solve inequality constraint

From Lagrange wiki, it says the Lagrangian method can only be used with equality constraint. If it is inequality constraint, can I add a slack variable to convert the inequality to equality constraint ...
0
votes
0answers
26 views

Slack variable required to convert inequality constraints if we are using Lagrangian method?

Most of the material I have read convert inequality constraint to equality constraint by adding slack variable then plug into augmented objective function with Lagrange multiplier Lambda. All of them ...
0
votes
1answer
14 views

Is the following linear in the choice variables?

I have a constrained optimization problem and need to show that the constraint is concave. Similar problem usually have linear constraints, so they don't need to worry about it. However, I am not ...
0
votes
1answer
43 views

Constraint optimisation problem

Given that $ \phi = lx+my+nz = 0$ and $ \psi = \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} - 1 = 0$, What are the max and min values of $ \kappa = \frac{x^2}{a^4} + \frac{y^2}{b^4} + \frac{z^...