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

arXiv: https://arxiv.org/abs/2609.30215 · HTML: https://mlsanity.com/paper/2609.30215/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 26 references that are in the map.
- First posted on arXiv 24 Sep 2026.

## Builds on (most cited first, 4 of 4)

- [Convex optimization using quantum oracles](https://mlsanity.com/paper/1809.00643/) (2018)
- [Quantum algorithms and lower bounds for convex optimization](https://mlsanity.com/paper/1809.01731/) (2018)
- [Reducing Isotropy and Volume to KLS: Faster Rounding and Volume Algorithms](https://mlsanity.com/paper/2008.02146/) (2020)
- [Closing the Oracle-Complexity Gap in Derivative-Free Convex Optimization: A Near-Quadratic Lower Bound from Exact Function Values](https://mlsanity.com/paper/2607.13335/) (2026)

## Built on by (most cited first, 1 of 1)

- [Lower Bounds for Parallel Diffusion Sampling](https://mlsanity.com/paper/2610.09166/) (2026)

## Abstract (arXiv)

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.