In 2010 he played on board 2 for team "Russia 2" at the 39th Chess Olympiad, held in Khanty-Mansiysk, Russia. His team finished sixth. He tied for first place with Sergey Karjakin, Peter Svidler, Dmitry Jakovenko, Dmitry Andreikin and Vladimir Potkin in the 2012 Russian Championship Superfinal. After the rapid playoff to determine the winner, Alekseev finished sixth. In the European Individual Chess Championship of 2013, he tied for first place with nine other players points and took the silver medal on tiebreak. In 2017, Alekseev won the championship of Saint Petersburg and the Viktor Korchnoi Memorial in St. Petersburg, this latter edging out Dmitry Kokarev, Gata Kamsky and Aleksandr Shimanov on tiebreak. In 2019, he won the 2nd Ferreira do Alentejo Open with a score of 7.5/9 points.