The contribution of this paper includes two aspects. First, we study the
lower bound complexity for the minimax optimization problem whose objective
function is the average of n individual smooth component functions. We
consider Proximal Incremental First-order (PIFO) algorithms which have access
to gradient and proximal oracle for each individual component. We develop a
novel approach for constructing adversarial problems, which partitions the
tridiagonal matrix of classical examples into n groups. This construction is
friendly to the analysis of incremental gradient and proximal oracle. With this
approach, we demonstrate the lower bounds of first-order algorithms for finding
an ε-suboptimal point and an ε-stationary point in
different settings. Second, we also derive the lower bounds of minimization
optimization with PIFO algorithms from our approach, which can cover the
results in \citep{woodworth2016tight} and improve the results in
\citep{zhou2019lower}