I'm not sure if the naval technology in Rome was adequate to navigate around Africa.
They probably would have preferred trading via the Suez.
Wasn't there an expedition of the Phoenicians sponsored by the Pharaohs to circumnavigate Africa? Assuming this is true (it is disputed) then there are at least technologies/techniques among the Empires citizens, which is the important bit.
Seeking India is a bit of a bugger. Perhaps it is interesting to the Western Empire, if there is a cost-saving in sailing around India rather than a potentially silted up Pharaohs Canal.
Alternatively, they could do it as a prestige project. Screw a library, I want to have a map of the world. Get me 15 ships, five exceptional cartographers, and send them out. Eastern Voyage, South-Eastern Voyage, South Western Voyage, Northern Voyage, and a Western Voyage.
The SW and SE Voyages bump into each other and go "Oh, cool, we've gone half-way each, trade maps, and get more accurate sailing information regarding travelling the other way.
The East gets decent maps of India and China (maybe Japan)
The Western Expedition eventually goes "Oh heeeeeellllll no" as they can't find anywhere to rest - unless their captains are brave as all hell/luck out and hit the Azores.
The Northern Expedition explores the Baltic Sea, comes back and asks to go North West, and depending on the bank balance of the sponsor, or their ability to find another, go North West, risk being destroyed, but find Greenland, and maybe Newfoundland.
But honestly, you need a wealthy man, with one hell of an ego, who REALLY likes maps. An admiral might do, an Emperor in a golden age (rather than a crisis period - prevent the crisis, perhaps they have the cash), or an Phoenician trading cabal that really want to get on good terms with the Emperor.