  Much recent research effort has been directed to the development of efficient
algorithms for solving minimax problems with theoretical convergence guarantees
due to the relevance of these problems to a few emergent applications. In this
paper, we propose a unified single-loop alternating gradient projection (AGP)
algorithm for solving smooth nonconvex-(strongly) concave and (strongly)
convex-nonconcave minimax problems. AGP employs simple gradient projection
steps for updating the primal and dual variables alternatively at each
iteration. We show that it can find an $\varepsilon$-stationary point of the
objective function in $\mathcal{O}\left( \varepsilon ^{-2} \right)$ (resp.
$\mathcal{O}\left( \varepsilon ^{-4} \right)$) iterations under
nonconvex-strongly concave (resp. nonconvex-concave) setting. Moreover, its
gradient complexity to obtain an $\varepsilon$-stationary point of the
objective function is bounded by $\mathcal{O}\left( \varepsilon ^{-2} \right)$
(resp., $\mathcal{O}\left( \varepsilon ^{-4} \right)$) under the strongly
convex-nonconcave (resp., convex-nonconcave) setting. To the best of our
knowledge, this is the first time that a simple and unified single-loop
algorithm is developed for solving both nonconvex-(strongly) concave and
(strongly) convex-nonconcave minimax problems. Moreover, the complexity results
for solving the latter (strongly) convex-nonconcave minimax problems have never
been obtained before in the literature. Numerical results show the efficiency
of the proposed AGP algorithm. Furthermore, we extend the AGP algorithm by
presenting a block alternating proximal gradient (BAPG) algorithm for solving
more general multi-block nonsmooth nonconvex-(strongly) concave and (strongly)
convex-nonconcave minimax problems. We can similarly establish the gradient
complexity of the proposed algorithm under these four different settings.
