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This paper presents a directional proximal point method (DPPM) to derive theminimum of any C1-smooth function f. The proposed method requires a functionpersistent a local convex segment along the descent direction at anynon-critical point (referred to a DLC direction at the point). The proposedDPPM can determine a DLC direction by solving a two-dimensional quadraticoptimization problem, regardless of the dimensionally of the functionvariables. Along that direction, the DPPM then updates by solving aone-dimensional optimization problem. This gives the DPPM advantage overcompetitive methods when dealing with large-scale problems, involving a largenumber 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 asingle critical point. For strongly convex quadratic functions, we demonstratethat the rate at which the error sequence converges to zero can beR-superlinear, regardless of the dimension of variables.

Unconstrained optimization using the directional proximal point method