AI-Assisted Discovery of Convex Relaxations via Dual Agents

• URL: http://arxiv.org/abs/2606.31182v1
• Date: Tue, 30 Jun 2026 06:10:25 GMT
• Title: AI-Assisted Discovery of Convex Relaxations via Dual Agents
• Abstract summary: We show that a lower bound holds for every admissible function and follows from a convex relaxation giving stronger bounds.<n>A theory verifies each agent one and searches for counterexamples, and every reported bound is certified by an explicit dual-feasible point in the interval.
• Score: 56.60366723277675
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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