Optimizing constants in mathematical inequalities has always been a challenge that faces limits at both ends: upper bounds are obtained through extremal constructions, while lower bounds require proofs valid for every admissible function. Traditionally, the search for convex relaxations that tighten these lower bounds relied on human intuition and tedious analytical verifications. However, a new paradigm is emerging where artificial intelligence and AI agents become active partners in this process.
Imagine a system composed of two complementary agents: a coding agent that, by exploring a space of constraints, proposes new candidate convex relations, and a theoretical agent that verifies each proposal, seeks counterexamples, and certifies validity through rigorous interval arithmetic. This dual-agent approach not only accelerates the discovery of tighter relaxations but also guarantees that each reported bound is supported by an explicit dual feasible point. Its effectiveness has been demonstrated by improving lower bounds of constants such as the autocorrelation inequality and Erdos' minimum overlap, raising them from 1.28 to 1.2937 and from 0.379005 to 0.37912, respectively.
Beyond pure mathematics, this dual-agent concept has direct applications in business. In custom software development, for example, workflows can be built where one agent generates optimization hypotheses (such as resource configurations or algorithm parameters) and another validates them in simulated or real environments. Companies like Q2BSTUDIO already integrate these principles into their artificial intelligence for businesses solutions, offering AI agents that automate both exploration and verification in complex tasks.
In the field of cybersecurity, one agent could propose patches or security configurations while another tests them against attack vectors. In business intelligence, tools like Power BI benefit from agents that generate hypotheses about trends and a second agent that cross-checks them against historical data, improving the quality of reports. Q2BSTUDIO offers business intelligence services that incorporate these dual-validation mechanisms to ensure decisions based on reliable data.
The infrastructure supporting these systems typically relies on AWS and Azure cloud services, where multiple agent instances with parallel computing capabilities are deployed. The company also develops custom applications that integrate convex optimization and interval verification modules, adapting to sectors such as logistics, finance, or energy. AWS and Azure cloud services provide the elasticity needed to run massive relaxation search experiments, while custom software ensures that each solution is aligned with business objectives.
This AI-assisted discovery approach marks a paradigm shift: we no longer depend solely on human reasoning to find new proofs or relaxations. AI agents collaborate in a continuous cycle of proposal and validation, similar to the scientific method but accelerated by orders of magnitude. Q2BSTUDIO, as a software development and technology company, is at the forefront of implementing these dual-agent architectures in real projects, helping its clients solve complex optimization problems, from improving mathematical bounds to optimizing business processes.
The integration of AI agents in the discovery of convex relaxations not only has academic implications but also opens the door to new analysis and design tools in engineering, economics, and operations research. With the support of solutions like those offered by Q2BSTUDIO, companies can leverage this technology to gain competitive advantages, whether through custom applications that incorporate dual verification logic or through cloud platforms that scale experimentation. The future of optimization is collaborative, and dual agents are its protagonists.
