If we want to find the local minima of a function $p(x)$, $x∈R^{2n}$, I understand we can use the gradient descent method: $x_{k+1}=x_k−α_k ∇_{x}p(x)$ where $α_k$ is the step size and $∇_{x} p(x)$ is the gradient of $p(x)$.
Moreover, I understand one way of solving this problem is to compute the gradient, $g_{k}=∇_{x}p(x)$, project it onto the tangent plane, $h_{k}=gk−(g_{k}⋅x_{k})x_{k}$, and normalize it, $n_{k}=h_{k}/|h_{k}|$. Consequently, one can use $x_{k+1}=x_{k}cosϕ_{k}+n_{k}sinϕ_{k} $ and perform a one-dimensional search for $ϕ_{k}$ to obtain a solution that is constrained to a sphere i.e. $||x||=1$ (see response by joriki on Gradient Descent with constraints).
My questions are:
1) what if $x$ is so large that one divides it into subvectors $x_{1}, x_{2}∈R^{n}$, can we still find a local minima of $p(x)$ on a sphere (i.e. $||x||=1$) using any gradient descent method?
2) Is there a way of solving this with the condition $0<||x||< 1$?
Any suggestions will be much appreciated?
Thanks.
