High-precision linear minimization is no slower than projection

Abstract

This note demonstrates that, for all compact convex sets, high-precision linear minimization can be performed via a single evaluation of the projection and a scalar-vector multiplication. In consequence, if $\epsilon$-approximate linear minimization takes at least $L(\epsilon )$ real vector-arithmetic operations and projection requires $P$ operations, then $\mathcal{O}(P)\ge \mathcal{O}(L(\epsilon ))$ is guaranteed. This concept is expounded with examples, an explicit error bound, and an exact linear minimization result for polyhedral sets.

7 pages, 1 figure