Finding the minimum difference between each element of one vector and another vector

I would use a step function sorted on the first vector. This will avoid loops and is pretty fast in R.

x <- rnorm(1000)
y <- rnorm(1000)
sorted.x <- sort(x)
myfun <- stepfun(sorted.x, 0:length(x))

Now myfun(1) will give you the index of the largest element of sorted.x whose value is less than 1. In my case,

> myfun(1)  
[1] 842
> sorted.x[842]
[1] 0.997574
> sorted.x[843]
[1] 1.014771

So you know that the closest element is either sorted.x[myfun(1)] or sorted.x[myfun(1) + 1]. Consequently (and padding for 0),

indices <- pmin(pmax(1, myfun(y)), length(sorted.x) - 1)
mindist <- pmin(abs(y - sorted.x[indices]), abs(y - sorted.x[indices + 1]))

start by sorting obj1

then you can do a binary search in obj1 for each element of obj2. knowing where the element would be, you can compare the distance to the two nearby elements of obj1, giving you the minimum distance.

runtime (where n1 = |obj1| and n2 = |obj2|): (n1 + n2) log (n1)