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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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Linearize if-then constraints

For continuous variables $x$ and $y$, the constraints are: ...
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Adding inequaltiy contrains in ODE's using bvp4c in Matlab

Consider following code in Matlab: ...
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(...
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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 ...
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 $\dot{x} = f\left( x, u \right)$...
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 ...
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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 ...
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 ...
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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 ...
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 ...
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 ...
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 ...
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 ...
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. ...
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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 ...
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 $\mathcal E$-...
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 ...
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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!
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 ...