"promote a solution" is correct and usable in written English.
You can use it whenever you want to talk about encouraging or advocating for a problem to be solved. For example, "We need to promote a solution to combat climate change."
The added water was found to play an important role to promote a solution-precipitation process inside the calcium powder compact, allowing low temperature sintering to form dense calcium carbonate.
Now a San Francisco company, CleanPath Ventures, is promoting a solution to allow homeowners to keep their trees and go solar at the same time.
It is one that will require a real nation-building commitment, and a real effort to stabilize the region by simultaneously promoting a solution to the Israeli-Palestinian conflict.
'We must seize this opportunity to promote a peaceful solution,' the Yemeni ambassador declared.
The efforts of significant Syrian opposition groups, meeting in Rome at the end of July, to promote a peaceful solution (in the Sant'Egidio statement), were barely reported.
Both leaders urged continued support of a peace plan negotiated by Mr. Annan "to promote a political solution to the problem in Syria".
Hong also praised Annan's efforts and said China is open "towards any proposals that would help promote a political solution to the Syria issue".
Special Envoy Annan's mediation efforts and the U.N. monitoring mission, to promote a political solution to the problem in Syria".
"This attack came at a time when more effective, prominent circles were prepared to promote a democratic solution to the Kurdish issue at large," said Umit Firat, a Kurdish intellectual.
His mission, he said, was "to seek an urgent end to all violence and human rights violations, and to initiate the effort to promote a peaceful solution to the Syrian crisis".
