What is the best way to clean outside bottoms of pots and pans?

Beekeepers Friend. It is similar to scouring powders like Comet, but is less abrasive, so it won't scratch the finish.

bicarbonate of soda plus white vinegar

The BEST is Bar Keepers Friend in the powder form. It is good on copper bottoms as well as stainless. I have used this for years and my pots look always like pretty close to new. I am quite sure you will be very happy with this product. If heavy soiled you may have to do 2 x. After that do this every time you use your pots inside and out.

I have had the best luck with a grit based cleaner such as Bar Keeper's Best Friend and some steel wool and some elbow grease. It just takes time and work.