I want to find an $n$-vector $\hat{\theta}$ that maximizes a function $f(\theta)$ subject to the $p$-norm constraint $||\theta||_p = c$.
Is there a general parametrization of $p$-norm hyperspheres that makes this easy?
I want to find an $n$-vector $\hat{\theta}$ that maximizes a function $f(\theta)$ subject to the $p$-norm constraint $||\theta||_p = c$.
Is there a general parametrization of $p$-norm hyperspheres that makes this easy?
The standard 2-norm hypersphere is parametrized by hyperspherical coordinates, which you can easily turn into a parametrization of the $p$-norm hypersphere by transforming each coordinate as $x_i \mapsto \operatorname{sgn}(x_i) \lvert x_i \rvert^{2/p}$.
However, I would not recommend using this for solving an optimization problem, because the parametrization is singular at the poles, and at many other points when $p \le 1$ or $p = \infty$. You should look into Lagrange multipliers instead.
a smooth surface.
– 2011-03-10