  For a class of discrete quasi convex functions called semi-strictly quasi
M$^\natural$-convex functions, we investigate fundamental issues relating to
minimization, such as optimality condition by local optimality, minimizer cut
property, geodesic property, and proximity property. Emphasis is put on
comparisons with (usual) M$^\natural$-convex functions. The same optimality
condition and a weaker form of the minimizer cut property hold for
semi-strictly quasi M$^\natural$-convex functions, while geodesic property and
proximity property fail.
