can we find the optimize value of a variable on which two other variables depend

I have two vectors x and y whose values are calculated for different values of third parameter z. when I plotted x and y against z, both x and y are linearly rising(almost parallel) for certain values of z and after that x behaves non linearly whereas y is linear for all values of z. My question is : can I find the value of z for which ratio x/y has optimum value (high)(x should be expected to have value as high as possible whereas y should required to be low). do I need to apply linear programming or some optimization method/technique. here we can choose any relation between x and y (e.g adding weights to x and y)as an objective function in case we apply any optimization method/technique. further I have to apply the same criteria on all the rows of a matrix to calculate a value /range of x please help

[Walter Roberson]

Is the non-linear portion monotonically increasing? Is it faster than linear in that section? When you say that you have two vectors, do you mean that you have two formula, or do you mean that you have two vectors of experimental numeric values?

If you have two vectors of experimental numeric values then you can just calculate

[~, maxidx] = max(x ./ y);

and then x(maxidx) and y(maxidx)

[studentambitious]

1. The non linear portion is not monotonically increasing instead it increases or decreases randomly for values of z.
2. the two vectors are of numerically experimental numeric values, but max(x./y) always shows the maximum value for z=1, as at that point x has some value but y is nearly zero. the value of x at z=1, is very less to be accepted.

[Walter Roberson]

Any non-zero weights you might put on to x and y would be the same as a constant linear scaling, so you would always end up with the same location. I do not understand what is being optimized?