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I haven't done this in quite a while so excuse my perhaps silly question.

I'm looking for a solution to a minimization problem (if there is one), that goes like this:

I want to minimize (global) $f(x,y,z)=x^2+y^2+z^2$ under the condition that $ax+by+cz = 1$ with a,b,c being constants.

Is there even a solution to this problem, since there are two equations and 3 unknowns? What if I have these two equations with lets say 20 unknown variables?

It would be nice if anyone could either post the solution to this problem, or some hints (=> google search terms) so that i can lookup this problem.

But before all: is there a single solution to this problem?

thanks.

egg
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  • Do you know Lagrange multipliers ? – Claude Leibovici Jul 15 '14 at 08:12
  • First, you should think geometrically. Your function $f$ corresponds to the euclidian norm in $\mathbb{R}^3$ and your equation $ax+by+cz=1$ is a plane, draw a figure to see what exactly you're trying to get. Then for optimization under constraints, there is the classical KKT theory. – user37238 Jul 15 '14 at 08:14

2 Answers2

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Hint

If you know Lagrange multipliers, then you need to minimize $$F(x,y,z,\lambda )=x^2+y^2+z^2+\lambda (ax+by+cz-1)$$ So, compute $F'_x,F'_y,F'_z,F'_{\lambda}$, set each of them to be equal to $0$; so, you get a linear system of four equations for four unknowns, the solution of which being pretty trivial.

As you see, you could have as many variables as desired.

I am sure that you can take from here.

Claude Leibovici
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There's a geometric interpretation here. You're minimizing the norm of vectors on a hyperplane. You can do this by computing the distance of the plane from the origin, using geometric arguments. This is of course equivalent to solving the minimization problem using KKT multipliers. (Sorry can't comment till I get more rep so I thought I'd put it in as an answer).

wonko
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