AI-Assisted Discovery of Convex Relaxations via Dual Agents

2026-06-30

AI Summary

This work proposes a dual-agent collaborative framework for automatically discovering convex relaxations to strengthen lower bounds in nonconvex optimization problems. An encoding agent generates tight constraints, while a theory agent validates their correctness through explicit dual feasible points and rigorous interval arithmetic. The approach pioneers the integration of large language model–driven autonomous research paradigms into convex relaxation construction, unifying automated lower-bound optimization with formal mathematical proof. The method achieves new state-of-the-art results on two classical optimization constants: improving $C_{6.2}$ from 1.28 to 1.2937 and $C_{6.5}$ from 0.379005 to 0.37912.

Abstract

Recent work shows that LLM agents can improve sharp-constant inequalities by searching for extremal constructions, which yield upper bounds. We address the complementary side: a lower bound holds for every admissible function and follows from a convex relaxation of the nonconvex problem, with tighter relaxations giving stronger bounds. We instantiate the autoresearch paradigm to discover such relaxations: a coding agent proposes valid tightening constraints, a theory agent verifies each one and searches for counterexamples, and every reported bound is certified by an explicit dual-feasible point checked in rigorous interval arithmetic. On two optimization constants studied by \citet{tao2025alphaevolve} - the first autocorrelation inequality ($C_{6.2}$) and the Erdős minimum-overlap constant ($C_{6.5}$) - we improve the certified lower bounds from $1.28$ to $1.2937$ and from $0.379005$ to $0.37912$, respectively.

Problem

• convex relaxation
• lower bounds
• nonconvex optimization
• dual feasibility
• extremal constructions

Innovation

• convex relaxation
• dual agents
• interval arithmetic
• lower bounds
• AI-assisted discovery