Complexity of optimizing over the integers

TL;DR

This paper introduces a unified framework for analyzing the complexity of optimization problems, bridging continuous and discrete cases, and applies it to mixed-integer convex optimization to foster cross-disciplinary understanding.

Contribution

It develops a general formalism for complexity in optimization, unifying continuous and discrete cases, and studies mixed-integer convex optimization within this framework.

Findings

Unified complexity framework for optimization problems

Analysis of mixed-integer convex optimization complexity

Bridging continuous and discrete optimization communities

Abstract

In the first part of this paper, we present a unified framework for analyzing the algorithmic complexity of any optimization problem, whether it be continuous or discrete in nature. This helps to formalize notions like "input", "size" and "complexity" in the context of general mathematical optimization, avoiding context dependent definitions which is one of the sources of difference in the treatment of complexity within continuous and discrete optimization. In the second part of the paper, we employ the language developed in the first part to study information theoretic and algorithmic complexity of {\em mixed-integer convex optimization}, which contains as a special case continuous convex optimization on the one hand and pure integer optimization on the other. We strive for the maximum possible generality in our exposition. We hope that this paper contains material that both continuous optimizers and discrete optimizers find new and interesting, even though almost all of the material presented is common knowledge in one or the other community. We see the main merit of this paper as bringing together all of this information under one unifying umbrella with the hope that this will act as yet another catalyst for more interaction across the continuous-discrete divide. In fact, our motivation behind Part I of the paper is to provide a common language for both communities.