GPT-5.6 Solves 30-Year Convex Optimization Lower Bound Using a Detailed Prompt

GPT-5.6 Solves 30-Year Convex Optimization Lower Bound Using a Detailed Prompt

AI‑Generated Proof Closes a 30‑Year Gap in Convex Optimization

Takeaway: GPT‑5.6, when supplied with a carefully crafted ten‑page prompt, produced a proof that the lower bound on the number of function evaluations needed to optimize convex, Lipschitz functions over a spherical domain is Ω(d²). This matches the runtime of a 30‑year‑old algorithm and therefore resolves a long‑standing open question in convex optimization.

The Result in Plain Terms

The proof shows that any algorithm that accesses a convex, Lipschitz function only through point evaluations must query the function at least on the order of d² times, where d is the dimension of the domain. This lower bound exactly matches the complexity of the best known algorithm from the early 1990s, meaning the algorithm is optimal up to constant factors.

"Showing upper bounds on time complexity is easy because it’s just the runtime of your algorithm. Showing (nontrivial) lower bounds is usually much harder because it requires constraining all algorithms. This proof apparently shows that the lower bound time complexity is equal to the time complexity of an existing 30‑year‑old algorithm: it requires Ω(d²) function evaluations to solve over this class of functions." – @alternator

Why the Prompt Matters

The author did not simply ask GPT‑5.6 to "solve the problem". Instead, they prepared a ten‑page prompt containing:

  1. A concise statement of the conjecture.
  2. A literature review of prior attempts and known techniques.
  3. Explicit instructions on how the model should explore candidate proofs, including which sub‑gradients and oracle constructions to consider.
  4. A set of concrete lemmas and auxiliary results that the model could reuse.

The prompt mirrors the style of OpenAI’s earlier Cyclic Double Cover (CDC) prompt, which also leveraged extensive domain knowledge to steer the model.

"The prompt is on page 27 here[1]. It is ten pages of advanced mathematics priming the model in the right direction, apparently informed by a year of prior research. That doesn’t invalidate the result if it is genuine, but it is worth noting that this wasn’t a matter of ‘ChatGPT, solve this unsolved problem. Make no mistakes.’ and required substantial domain expertise and human research beforehand." – @applfanboysbgon

Verification and Peer Review Status

The proof has not yet undergone formal peer review. The author formally verified the result in the Lean theorem prover, and the Lean check passed, providing an additional layer of confidence.

"Not yet peer reviewed" – @sdwvit

The community’s caution reflects a broader pattern: AI‑generated proofs must still be scrutinized for hidden gaps, just as human‑written proofs are.

Implications for Convex Optimization Theory

  1. Optimality of Classical Algorithms – The Ω(d²) lower bound confirms that the classic algorithm from the early 1990s cannot be asymptotically improved using only function‑value queries.
  2. Gradient Oracle Perspective – If a gradient oracle is available, a gradient can be approximated with d function evaluations, suggesting that the lower bound may be tight for gradient‑based methods as well, though a rigorous reduction remains open.
  3. Methodological Shift – The success of a prompt‑driven LLM approach demonstrates that large language models can now assist in theoretical breakthroughs, not just computational experiments.

Community Reactions

  • Skepticism about Novelty – Some commenters note that the result hinges heavily on the human‑crafted prompt and prior year‑long work, questioning how much of the intellectual contribution belongs to the model versus the author.

    "It’s not clear to me the degree to which ‘GPT‑5.6 used a prompt’ … or the author basically did all the work himself and assigned it to GPT‑5.6 out of enthusiasm." – @YeGoblynQueenne

  • Broader AI Impact – Others see this as a sign that low‑hanging research problems may soon be automated, pushing researchers toward more creative, open‑ended questions.

    "I don’t think researchers in math/TCS will be made obsolete, but I think it will instead no longer make sense to work on any low‑hanging, or even medium‑hanging fruit. We'll be needed for problems where actual novel approaches are needed." – @rakel_rakel

  • Technical Curiosity – A few participants asked about the underlying architecture (Sol Pro vs. Ultra) and whether multi‑agent systems contributed to the result, though concrete details remain scarce.

    "My understanding is that ChatGPT Pro is effectively a multi‑agent system… Ultra is more similar to Claude‑Code UltraCode where the main agent can choose to create a dynamic JS workflow…" – @d4rkp4ttern

Limitations and Open Questions

  • Formal Peer Review – The community awaits a traditional peer‑reviewed publication to confirm the correctness of the proof.
  • Generality of the Prompt Technique – It is unclear whether similar prompts can solve other longstanding conjectures (e.g., the abc conjecture) without comparable domain expertise.
  • Model Access to Prior Chats – The extent to which GPT‑5.6 leveraged memory of earlier interactions (GPT‑5.4/5.5 sessions) is not publicly known, raising questions about reproducibility.

Bottom Line

GPT‑5.6, guided by an extensive, expert‑level prompt, succeeded in producing a formally verified proof that settles a 30‑year‑old lower‑bound question in convex optimization. While the result is promising, it remains unreviewed, and the broader significance hinges on how reproducible and generalizable this prompt‑driven methodology proves to be.

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