Making a plot of an interpolation also requires that you give a variable. Examples of this are given in the documentation:

f = Interpolation[{1, 2, 3, 5, 8, 5}]

Plot[f[x], {x, 1, 6}]

We can follow the same pattern with NMinmize. Because we are using an interpolation, I have included constraints for NMinimize to constrain it to the region of the iterpolation. In this case the interpolation is where the independant variable is between -1 and 1.

data={{-1., 0.175194}, {-0.9, -0.026003}, {-0.8, -0.0807706}, {-0.7, -0.142006}, {-0.6, -0.250069}, {-0.5, -0.262721}, {-0.4, -0.271475}, {-0.3, -0.196399}, {-0.2, -0.301731}, {-0.1, -0.0954844}, {5.55112*10^-17, -0.0656419}, {0.1, 0.0820605}, {0.2, 0.204884}, {0.3, 0.420364}, {0.4, 0.551297}, {0.5, 0.74569}, {0.6, 0.837153}, {0.7, 1.14175}, {0.8, 1.3972}, {0.9, 1.59105}, {1., 1.73636}};

f = Interpolation[data];

NMinimize[{f[x],-1<x<1}, x]

Minimizing an interpolation doesn't appear yet as a suggested operation. I am not sure however it is a common task. With real world data, you usually see optimization done on statistical models or something similar. It's important to remember that the minimum can depend heavily on the kind of interpolation that is used and for sufficiently smooth interpolations, it really just gives the rough location of the smallest element. If you would like to suggest improvements on the predictive interface, you can always do so by clicking orange speech bubble that appears to the right of the suggestion bar.