using System;

using System.Collections.Generic;

using System.Linq;

using System.Xml.Linq;

public class Program

{

    public static void Main()

    {

        const string XML = @"

<root>

    <Payment>

        <PaymentId>1</PaymentId>

        <Total>50</Total>        

        <Invoice>

            <Id>555</Id>

            <Amount>30</Amount>

        </Invoice>

        <Invoice>

            <Id>111</Id>

            <Amount>30</Amount>

        </Invoice>

    </Payment>

    <Payment>

        <PaymentId>2</PaymentId>

        <Total>20</Total>

        <Invoice>

            <Id>222</Id>

            <Amount>20</Amount>

        </Invoice>

    </Payment>

    <Payment>

        <PaymentId>3</PaymentId>

        <Total>80</Total>

        <Invoice>

            <Id>888</Id>

            <Amount>80</Amount>

        </Invoice>

    </Payment>

</root>

";

​

            XElement xml = XElement.Parse(XML);

            List<string> ids = new List<string> { "111", "222", "333" };

            var result = xml

                .Elements("Payment")

                .Where(p => {

                    var invoiceIds = p.Elements("Invoice").Elements("Id").Select(o => o.Value);

                    return ids.Intersect(invoiceIds).Any();

                    }

                );

​

            foreach (var item in result)

            {

                Console.WriteLine("PaymentId: {0}", (string)item.Element("PaymentId"));

            }

        }

}