  This paper presents a stochastic approximation proximal subgradient (SAPS)
method for stochastic convex-concave minimax optimization. By accessing
unbiased and variance bounded approximate subgradients, we show that this
algorithm exhibits ${\rm O}(N^{-1/2})$ expected convergence rate of the minimax
optimality measure if the parameters in the algorithm are properly chosen,
where $N$ denotes the number of iterations. Moreover, we show that the
algorithm has ${\rm O}(\log(N)N^{-1/2})$ minimax optimality measure bound with
high probability. Further we study a specific stochastic convex-concave minimax
optimization problems arising from stochastic convex conic optimization
problems, which the the bounded subgradient condition is fail. To overcome the
lack of the bounded subgradient conditions in convex-concave minimax problems,
we propose a linearized stochastic approximation augmented Lagrange (LSAAL)
method and prove that this algorithm exhibits ${\rm O}(N^{-1/2})$ expected
convergence rate for the minimax optimality measure and ${\rm
O}(\log^2(N)N^{-1/2})$ minimax optimality measure bound with high probability
as well. Preliminary numerical results demonstrate the effect of the SAPS and
LSAAL methods.
