# Questions tagged [constraints]

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

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### How to set up Lagrangian for matrix constraints?

Suppose we have a function $f: \mathbb{R} \to \mathbb{R}$ which we want to optimize subject to some constraint $g(x) \le c$, where $g:\mathbb{R} \to \mathbb{R}$. What we do is that we can set up a ...
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### Write logical operator all(x<a) in terms of Heavyside functions

I am currently solving a complex optimisation problem, with constraints that take the form: $1 - all(g(x)<a) <= 0$, meaning I require all values $g(x)$ (for some function $g$) to be below some ...
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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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### Optimize monotonic function in calculus of variations

I'm interested in the variational problem $$\min_{y} \int_a^b F(x,y(x),y'(x))dx \qquad \text{subject to} \quad -y'(x)\leq 0 \quad \forall x \tag{1}$$ i.e. $y(x)$ has to be monotonic. I ...
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### Importance of the Klee-Minty Cube in Optimization

Has anyone ever heard of the Klee-Minty Cube in Optimization? Supposedly, the Klee-Minty Cube shows the "flaws" of the Dantzig's Simplex Algorithm. Supposedly, Dantzig's Simplex Algorithm ...
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### Why does the Lagrange multiplier $\lambda$ change when the equality constraint is scaled?

Consider the problem $$\begin{array}{ll} \text{maximize} & x^2+y^2 \\ \text{subject to} & \dfrac{x^2}{25} + \dfrac{y^2}{9} = 1\end{array}$$ Solving this using the Lagrange multiplier method,...
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### How do you find redundant constraints for a feasible region?

I've found a few papers that deal with removing redundant inequality constraints for linear programs, but I'm only trying to find the non-redundant constraints that define a feasible region (i.e. I ...
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### ${L}_{1}$ (L1) Norm Regularized Minimization with of Convex Function with Linear Equality Constraint Using ADMM Framework

In section 6.3 of Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers there is a method for minimizing a loss function with l1 regularization. i.e. ...
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### Help needed to define a constraint in an optimization problem?

Given objective function is \begin{align} \underset{\mathbf{p},\mathbf{q}}{\text{min}}\hspace{4mm} (\mathbf{p*q})^T \mathbf{A}(\mathbf{p*q}) \hspace{4mm} \\ s.t \hspace{4mm}\mathbf{p^Te_p}-1=0\\\...
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