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I have a normal vector $n$ in a $d$ (large) dimensional space. This normal vector defines a hyperspace which crosses the origin. The points on the plane can be defined as $x$ where:

$$ x \cdot n = 0 $$

I would like to find $P$, which is the farthest point of intersection of the positive side of the unit cube and the hyperplane. Which means, I am looking for a $P$, where:

$$ \forall i \in [1 \dots d] : P_i \geq 0 $$ $$ \max_{i \in [1 \dots d]} P_i = 1 $$ $$ P \cdot n = 0 $$

It is trivial to solve this equation for $d=2$, but it gets complicated for higher dimensions. I tried to find an equation for the cube-hyperplane intersection, but I am not sure if I am on the right track.

How is it possible to solve a problem like this?

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  • $\begingroup$ Please check the statement of the problem. If $d=2$ and $n=(1,1)$ the feasible set is empty. $\endgroup$ – Christian Blatter Jul 11 '18 at 18:26
  • $\begingroup$ Yes, there are some rare cases, when there is no solution $\endgroup$ – Iter Ator Jul 11 '18 at 20:04
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Not sure if this will be easier to do, but here is something you could try:

You would like to maximize the (square of the) distance function $f:\mathbb{R}^d\to\mathbb{R}$, $$f(P_1,\dots,P_d) := d(\vec{P},0)^2= \sum_{i=1}^d P_i^2$$ subject to the following constraints:

$\sum_{i=1}^d P_in_i =0 \ \ \ \ $ (this ensures that the maximizing point is on the plane $x\cdot n=0$)

$0\leq P_i\leq 1$ for all $i=1,\dots,d \ \ \ \ $ (this ensures that the point is in the unit cube)

$\prod_{i=1}^d(P_i-1)=0 \ \ \ \ $ (this ensures that at least one of the $P_i$ is equal to 1)

In order to solve this maximization problem you could try to use the Kuhn-Tucker conditions. Best of luck!

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