Suppose we have an optimization problem $$ \mathbf{x} = (x_1, x_2, \ldots, x_m) = \arg\!\min_{\mathbf{x}\in \mathbb{R}^m}f(\mathbf{x})$$ and a second related problem: $$ \mathbf{y} = (y_1, y_2, \ldots, y_n) = \arg\!\min_{\mathbf{y}\in \mathbb{R}^n}g(\mathbf{y})$$ because we have the relations $$ \mathbf{x} = \mathbf{t}(\mathbf{y}) $$ and $$ f(\mathbf{t}(\mathbf{y})) = g(\mathbf{y}) $$ and $n>m$.
My question is: what is the disadvantage of solving the second problem instead of the first problem if you use a numerical optimizer?
An example: take for example a function $g(\mathbf{y})$ that only depends on $$x_1 = y_1 + y_2$$ $$x_2 = y_1+y_3$$ $$x_3 = y_1 + y_4$$ and $$x_4 =y_1+y_5$$. So minimizing $g(\mathbf{y})$ is equivalent with minimizing $$ f(\mathbf{x}) = g(0,x_1,x_2,x_3,x_4) $$
Is it harmful to solve the optimization problem the larger problem? Is it possible that a numerical optimizer has more troubles solving the larger problem, even though it doesn't matter which of all the extra solutions it gives?
edit: I use the Nelder-Mead algorithm.
update: in some cases the optimizer indicates the problem of a degenerate Nelder mead simplex . Is the root of the problem to be found in the extra variable?