凸関数の片側方向導関数と劣勾配

概要

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In general, for a convex function f : R^n → R ∪ {-∞, ∞} and any point (x, f(x)) in the graph G_f ⊂ R^<n+1>, it is possible that any lines or hyperplanes uniquely having (x, f(x)) in common with G_f may not be unique, or any line or hyperplane tangent to G_f and passing through (x, f(x)) may have, in G_f, other different common points than (x, f(x)). Accordingly, to solve optimal problems, it is necessary to consider differential properties of convex functions under a more general framework not necessarily assuming differentiability conditions. In this point, the important concepts are one-sided directional derivatives and subgradients. This paper, first, makes a study on the properties of one-sided directional derivatives and subgradients of convex functions, and second, on the relationships between these two concepts.
関西学院大学の論文