optimization

Closing the Oracle-Complexity Gap in Derivative-Free Convex Optimization: A Near-Quadratic Lower Bound from Exact Function Values

arXiv:2607.13335

summary

The paper proves a near‑quadratic lower bound on the number of exact function‑value queries needed to minimize a convex Lipschitz function over a Euclidean ball, closing a long‑standing gap, and extends the result to mixed‑integer convex optimization.

Abstract

We study the deterministic query complexity of minimizing a convex Lipschitz function over a $d$ -dimensional Euclidean ball using only exact function values. At accuracy $I^ (d^{- 1/2})$ , the previously applicable lower bound was $I^c ◯ (d)$ , inherited from the stronger full first-order oracle, while an upper bound from Protasov's value-only method requires $O (d^{2} lo g^{2} d)$ evaluations. By providing a lower bound of $I^c ◯ (\frac{d ^{2}}{l o g ( d + 1 )})$ on the oracle complexity in this setting, we thereby close this gap dating back to 1996, up to polylogarithmic factors. Furthermore, we are able to lift this result to the mixed-integer setting: Mixed-integer convex optimization with $d$ continuous and $n$ discrete variables using function values requires $I^~ c ◯ (d^{2} \cdot 2^{n})$ queries.

36 pages, 0 figures

Topics & keywords

#convex optimization #derivative-free optimization #oracle complexity #lower bounds #mixed-integer optimizationconvex Lipschitz functiondeterministic query complexityfunction-value oraclenear-quadratic lower boundmixed-integer convex optimization