Centerpoints: A link between optimization and convex geometry

TL;DR

This paper introduces a generalized centerpoint concept, develops an oracle-based algorithm for convex mixed-integer optimization, and demonstrates its optimality and structural properties, advancing understanding of optimization complexity.

Contribution

It generalizes centerpoints, links them to optimization algorithms, and proves their optimality and structural characteristics in convex mixed-integer problems.

Findings

Algorithms based on centerpoints are optimal in a certain sense.

Provides efficient algorithms for computing generalized centerpoints.

Establishes structural results about the centerpoint concept.

Abstract

We introduce a concept that generalizes several different notions of a "centerpoint" in the literature. We develop an oracle-based algorithm for convex mixed-integer optimization based on centerpoints. Further, we show that algorithms based on centerpoints are "best possible" in a certain sense. Motivated by this, we establish several structural results about this concept and provide efficient algorithms for computing these points. Our main motivation is to understand the complexity of oracle based convex mixed-integer optimization.