Information Complexity of Mixed-integer Convex Optimization

TL;DR

This paper explores the fundamental limits of mixed-integer convex optimization, establishing new lower bounds, transferring results from continuous to mixed-integer settings, and analyzing the impact of partial information oracles on algorithm efficiency.

Contribution

It introduces a transfer principle for lower bounds from continuous to mixed-integer convex optimization and studies the effects of partial first-order oracles on information complexity.

Findings

Established new lower bounds for first-order oracles in mixed-integer convex optimization.

Developed algorithms that operate under less informative, partial oracles.

Proved that partial oracles can require more iterations than full first-order oracles.

Abstract

We investigate the information complexity of mixed-integer convex optimization under different types of oracles. We establish new lower bounds for the standard first-order oracle, improving upon the previous best known lower bound. This leaves only a lower order linear term (in the dimension) as the gap between the lower and upper bounds. This is derived as a corollary of a more fundamental ``transfer" result that shows how lower bounds on information complexity of continuous convex optimization under different oracles can be transferred to the mixed-integer setting in a black-box manner. Further, we (to the best of our knowledge) initiate the study of, and obtain the first set of results on, information complexity under oracles that only reveal \emph{partial} first-order information, e.g., where one can only make a binary query over the function value or subgradient at a given point. We give algorithms for (mixed-integer) convex optimization that work under these less informative oracles. We also give lower bounds showing that, for some of these oracles, every algorithm requires more iterations to achieve a target error compared to when complete first-order information is available. That is, these oracles are provably less informative than full first-order oracles for the purpose of optimization.