# How do exponential size solutions arise in semidefinite programming?

arXiv: https://arxiv.org/abs/2103.00041 · HTML: https://mlsanity.com/paper/2103.00041/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Feb 2021.

## Built on by (most cited first, 2 of 2)

- [On Classifying Continuous Constraint Satisfaction Problems](https://mlsanity.com/paper/2106.02397/) (2021)
- [Learning quantum Hamiltonians at any temperature in polynomial time with Chebyshev and bit complexity](https://mlsanity.com/paper/2402.05552/) (2024)

## Abstract (arXiv)

A striking pathology of semidefinite programs (SDPs) is illustrated by a classical example of Khachiyan: feasible solutions in SDPs may need exponential space even to write down. Such exponential size solutions are the main obstacle to solve a long standing, fundamental open problem: can we decide feasibility of SDPs in polynomial time? The consensus seems that SDPs with large size solutions are rare. However, here we prove that they are actually quite common: a linear change of variables transforms every strictly feasible SDP into a Khachiyan type SDP, in which the leading variables are large. As to ``how large", that depends on the singularity degree of a dual problem. Further, we present some SDPs coming from sum-of-squares proofs, in which large solutions appear naturally, without any change of variables. We also partially answer the question: how do we represent such large solutions in polynomial space?