  In this paper, we discuss variational inequality (VI) problems without
monotonicity from the perspective of convergence of projection-type algorithms.
In particular, we identify existing conditions as well as present new
conditions that are sufficient to guarantee convergence. The first half of the
paper focuses on the case where a Minty solution exists (also known as Minty
condition), which is a common assumption in the recent developments for
non-monotone VI. The second half explores alternative sufficient conditions
that are different from the existing ones such as monotonicity or Minty
condition, using an algorithm-based approach. Through examples and convergence
analysis, we show that these conditions are capable of characterizing different
classes of VI problems where the algorithms are guaranteed to converge.
