# Questions tagged [lagrange-multiplier]

This tag is for the questions on Lagrange multipliers. The method of Lagrange multipliers (named after Joseph Louis Lagrange) provides a strategy for finding the local maxima and minima of a function subject to equality constraints.

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### How to solve second-order objective function with orthogonal constraint

I want to minimize $tr(P^TAP-P^TB)$ with the constraint $P^TP=I$. $P, A, B$ are all matrices, $I$ is the identity matrix, $tr$ means the trace of a matrix. P: size(m,l) A: size(m,m) non-symmetric ...
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### Lagrange multipliers for an optimization problem

Consider the nonlinear program $$\underset{\mathbb{R}^2}{\text{min}} f(x_1 , x_2 ) = x_1^2 - x_1 x_2 + x_2^2 - x_1 - x_2$$ subject to the conditions: ...
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### Minimizing $f(x,y,z)$ subject to a quadratic constraint

Given that $C(x,y,z) = x^2 + 2 y^2 + 3 z^2 + x y + 2 x z + 3 y z + 5 x + 7 y + 10 z - 100 = 0$ Find the minimum of $f(x,y,z) = 2 x^2 + 3 y^2 + z^2 + 5 x y + 2 x z + 4 y z + x + 2 y + z$ My Attempt: ...
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### Minimizing $f(x,y,z)$ subject to a linear constraint

If $x + 2 y + 4 z= 10$, then what is the minimum of $$f(x,y,z) = x^2 + 2 y^2 + 3 z^2 + x y + 2 x z + 3 y z + 7 z$$ And at what $(x,y,z)$ does this occur ? My attempt: Using Lagrange's multiplier ...
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### Writing the Lagrange of a Normal Distribution

I am watching this Youtube video on how to derive the Multivariate Normal Distribution using the Principle of Maximum Entropy (https://www.youtube.com/watch?v=7qsB8ElrCC4 @ 3:13). Here, the Lagrange ...
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### analytical solution to a toy-sized but interesting optimization problem

My question follows after the initial context. My research includes solving a subproblem many many times - maximizing the Euclidean distance between a pair of convex neighborhoods, e.g. between an ...
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### Prove that the point on one circle farthest from another circle must lie on the line connecting centers.

This question comes from a section in a text book about Lagrange Multipliers. The context of the question is such however, that it is clearly intended that the solution be based upon reasoning related ...
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### Simplifying Minimization on $(x-y)^2-z^2=1$ : Possible?

Let $A=\left\{(x, y, z) \in \mathbb{R}^3 \mid(x-y)^2-z^2=1\right\}$, and I want to find all points in $A$ that are nearest to $(0,0,0)$. I tried to use Lagrange's method: \mathcal{L}(...
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