A Nearly Quadratic Lower Bound for Linear Optimization over Convex Bodies in the Membership Oracle Model

2026-09-24

AI Summary

This study investigates the randomized algorithmic complexity lower bounds for linear optimization, uniform sampling, and volume estimation of convex bodies within the membership query model. By constructing specific hard instances, it establishes near-quadratic time lower bounds for these problems. The core contribution lies in elevating the lower bound for uniform sampling from linear to near-quadratic for the first time, while simultaneously deriving an identical bound for volume estimation, thereby filling a notable theoretical gap. Furthermore, the obtained lower bound for linear optimization matches the best known upper bound up to logarithmic factors. These results significantly advance existing theoretical bounds and precisely delineate the fundamental complexity limits of the aforementioned computational tasks.

Abstract

We prove nearly quadratic lower bounds for randomized algorithms for linear optimization and uniform sampling over convex bodies in the membership oracle model. For linear optimization, this matches the known nearly quadratic upper bound up to a polylog factor in the dimension. For uniform sampling, this improves on the previous linear lower bound. Our construction also implies the same lower bound for volume estimation.

Problem

• linear optimization
• uniform sampling
• convex bodies
• membership oracle model
• lower bound

Innovation

• Membership Oracle Model
• Linear Optimization
• Uniform Sampling
• Convex Bodies
• Lower Bound