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Oracle-Complexity Gap in Derivative-Free Convex Optimization

Algorithms & optimization · posed by Vladimir Protasov (gap since 1996), 1996 · solved

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Statement

For deterministically minimizing a convex 1-Lipschitz function on the dd-dimensional ball using only exact function values, the query complexity sat between Ω(d)\Omega(d) and O(d2log⁡2d)O(d^2 \log^2 d) since 1996. The paper proves a near-quadratic lower bound Ω(d2/log⁡(d+1))\Omega(d^2 / \log(d+1)), closing the gap: Q(d,∼d−1/2)=Θ(d2)Q(d, \sim d^{-1/2}) = \Theta(d^2), a polynomial separation from full first-order information.

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No person has examined this. Nothing here has been checked at all. say whether it holds →

  1. proof attempt · #1

    GPT-5.6 Sol Pro, with Phillip Kerger

    The record says a model found this and names the people who worked on it. No ProbXiv account is credited for it, and nobody has answered for it here.

    AI involvement
    ai discovered
    — the result was found by a model.

    Kerger reports that GPT-5.6 Sol Pro solved the problem rather than the author, following a workflow like OpenAI's Cycle Double Cover effort. It first proved a Ω~(d2)\tilde{\Omega}(d^2) lower bound at accuracy of order d−3d^{-3} (after ~148 minutes), which was then refined to the order-d−1/2d^{-1/2} result via a further ~230-minute run. The author verified the arguments by hand and takes full responsibility.

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