Let $\varphi:\mathbb R\to\mathbb R$ with $$\varphi(x)\xrightarrow{|x|\to\infty}0,$$ $d\in\mathbb N$ and $$f_d(h):=\prod_{i=1}^d\sum_{k\in\mathbb Z}\varphi(h_i-k)\;\;\;\text{for }x\in\mathbb R^d.$$ Given $x^{(1)},\ldots,x^{(k)}\in[0,1)^d$, I want to (numerically) minimize $$g_d(x):=\sum_{i=1}^kf_d\left(x-x^{(i)}\right)\;\;\;\text{for }x\in[0,1)^d.$$ (I'm using the gradient descent method for this.)
Now imagine that we replace $d$ by $2d$. Assume $x$ is the minimizer of $g_{2d}$. Does $x_1,\ldots,x_d$ minimize $g_d$?
Notice that $$f_{2d}(h)=f_d(h_1,\ldots,h_d)f_d(h_{d+1},\ldots,h_{2d})\;\;\;\text{for all }h\in\mathbb R^{2d}\tag1.$$
I doubt that the answer to the question above is positive, but what I would actually to achieve is to obtain a $x\in[0,1)^{2d}$ such that
- $x$;
- $x_1,\ldots,x_d$ and $x_{d+1},\ldots,x_{2d}$
are (at least approximate) minimizers of $g_{2d}$ and $g_d$, respectively. Is that possible?