  This paper presents a directional proximal point method (DPPM) to derive the
minimum of any C1-smooth function f. The proposed method requires a function
persistent a local convex segment along the descent direction at any
non-critical point (referred to a DLC direction at the point). The proposed
DPPM can determine a DLC direction by solving a two-dimensional quadratic
optimization problem, regardless of the dimensionally of the function
variables. Along that direction, the DPPM then updates by solving a
one-dimensional optimization problem. This gives the DPPM advantage over
competitive methods when dealing with large-scale problems, involving a large
number of variables. We show that the DPPM converges to critical points of f.
We also provide conditions under which the entire DPPM sequence converges to a
single critical point. For strongly convex quadratic functions, we demonstrate
that the rate at which the error sequence converges to zero can be
R-superlinear, regardless of the dimension of variables.
