Is it permissible to take someone else's shoes – which are worth no more than twenty riyals – in exchange for my shoes, which were lost and are worth six hundred riyals?

If something is stolen from you and you find it with the thief or someone else, you have the right to take it back, even by force, as long as it does not lead to a greater harm. It is not a condition to refer the matter to a judge. If you do not find the exact item that is rightfully yours, but you find money belonging to the usurper, it is permissible for you to take your right from it. Al-Bukhari and Ibn Sirin supported the permissibility for the wronged person to take retribution from the oppressor's wealth. Therefore, if you are certain with absolute conviction that the shoe you found is your shoe, based on clear signs and indications, then there is no blame in what you have taken.