The first step will be to find the type of diamond shape your loved one wants or else consider round or princess cut. Next is to set the minimum carat weight that the receiver may desire.
You can start taking these measures until you are able to find a diamond that your budget allows:
First step is to lower the Clarity to low as VS2 prior to making other compromises. Next is to lower the Color to H bringing down the price closer to budget set for affordable engagement ring and wedding set. Lastly, lower the Cut going to Very Good in round diamonds, and Good in fancy shapes. You can obviously think of some drop in the carat weight so that it is within your budget. A variation of 10% or less in carat weight is extremely hard to notice visually.
If you are still outside your budget after these measures, you can go all over the procedure once again with new threshold by additionally reducing Clarity and to SI1. Reducing the Color as low as J can be an option. There is one more option if you plan to set your diamond in yellow gold that is to go down a little more in color up to K. and to make more affordable engagement ring decrease the Cut to as low as Good in round diamonds, and Fair in fancy shapes.