API for Customer Order Lookup

I would like to look up the order IDs associated with a customer account, searching on customer information. Ideally, I would search customer name or phone number, and find all orders associated with that customer. It doesn't look like the APIs support this.

The only workaround that I see is Retrieving All Orders and then parsing through them to match Name/Phone within billingAddress or shippingAddress. This could be incredibly lengthy, given the amount of order volume. We won't necessarily know the modifiedBefore/After information to time-box and reduce the number of returned orders.

Any help would be greatly appreciated!