High-precision linear minimization is no slower than projection

TL;DR

This paper shows that high-precision linear minimization over convex sets can be achieved efficiently using a single projection, making it comparable in complexity to projection operations alone.

Contribution

It establishes that linear minimization at high precision can be performed with essentially the same complexity as projection, unifying these operations for convex sets.

Findings

High-precision linear minimization can be reduced to a projection plus scalar multiplication.

The complexity of linear minimization is comparable to that of projection operations.

Provides explicit error bounds and exact solutions for polyhedral sets.

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 $\varepsilon$-approximate linear minimization takes at least $L(\varepsilon)$ real vector-arithmetic operations and projection requires $P$ operations, then $\mathcal{O}(P)\geq \mathcal{O}(L(\varepsilon))$ is guaranteed. This concept is expounded with examples, an explicit error bound, and an exact linear minimization result for polyhedral sets.