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OP-1231OpenOptimization

An overview of nonlinear optimization · 2022-opt-18

Optimization Theory, Algorithms, and Complexity · ICM 2022: An overview of nonlinear optimization

MSC 90C · Mathematical programming

Inferred from the community field; finer classification has not been reviewed.

Source assessment · not reviewedAs of 2026-08-18Source & evidence ↓
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Problem statement

Consider the strictly convex quadratic minimization problem

min⁡x∈Rnq(x):=12xTAx−bTx, \min_{x\in\mathbb{R}^n} q(x):=\tfrac12 x^{\mathsf T}Ax-b^{\mathsf T}x,

where A∈Rn×nA\in\mathbb{R}^{n\times n} is symmetric positive definite and b∈Rnb\in\mathbb{R}^n. Let gk=∇q(xk)g_k=\nabla q(x_k), let the exact steepest-descent stepsize at xkx_k be

αkSD=arg⁡min⁡α∈Rq(xk−αgk), \alpha_k^{\mathrm{SD}}=\arg\min_{\alpha\in\mathbb{R}}q(x_k-\alpha g_k),

and define the cyclic steepest-descent method with cycle length m≥1m\ge 1 by

xk+1=xk−αkgk,αmℓ+i=αmℓ+1SD(ℓ=0,1,…, i=1,…,m). x_{k+1}=x_k-\alpha_k g_k,\qquad \alpha_{m\ell+i}=\alpha_{m\ell+1}^{\mathrm{SD}} \quad (\ell=0,1,\ldots,\ i=1,\ldots,m).

Determine whether, for every dimension nn and every such quadratic problem, the generated iterates converge RR-superlinearly to the minimizer whenever m≥(n+1)/2m\ge (n+1)/2, where RR-superlinear convergence means that there is a nonnegative sequence vkv_k with ∥xk−x∗∥≤vk\|x_k-x_*\|\le v_k and vk+1/vk→0v_{k+1}/v_k\to 0, with x∗x_* the unique minimizer of qq.

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Progress & source assessment

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Status: unresolved.

The supplied wording says “BB method,” but the source’s surrounding passage and the cited paper identify the intended problem as the cyclic steepest-descent (CSD) method: every exact steepest-descent stepsize is reused for a fixed cycle length. No citable result located through July 31, 2026 proves or disproves the stated R-superlinear convergence guarantee for every dimension and every symmetric positive-definite quadratic whenever m >= (n+1)/2.

The principal source is Yu-Hong Dai and Roger Fletcher, “On the asymptotic behaviour of some new gradient methods,” Mathematical Programming 103 (2005), 541–559, https://doi.org/10.1007/s10107-004-0516-9. It analyzes the asymptotic behavior of the CSD method and reports that it is likely to be R-superlinearly convergent under the cycle-length condition, but this is evidence for a conjecture rather than a proof of the all-dimensional assertion.

A later partial result is Ya Shen, Qing-Na Li, and Yu-Hong Dai, “On the acceleration of gradient methods: the triangle steepest descent method,” arXiv:2501.16731 (version 1, 2025), https://arxiv.org/abs/2501.16731. Its CSD theorem proves that, for a two-dimensional strictly convex quadratic and any cycle length j >= 2, a subsequence of the gradients at cycle boundaries is R-superlinearly convergent. This does not establish the statement’s all-dimensional claim, nor does it state a complete result for the full iterate sequence under m >= (n+1)/2.

Related work includes A. Friedlander, J. M. Martínez, B. Molina, and M. Raydan, “Gradient Method with Retards and Generalizations,” SIAM Journal on Numerical Analysis 36 (1998), 275–289, https://doi.org/10.1137/S003614299427315X, which places CSD in a broader delayed-stepsize framework and proves global convergence under general conditions, but not the stated R-superlinear result; and Yu-Hong Dai, William W. Hager, Klaus Schittkowski, and Hongchao Zhang, “The cyclic Barzilai–Borwein method for unconstrained optimization,” IMA Journal of Numerical Analysis 26 (2006), 604–627, https://doi.org/10.1093/imanum/drl006, which studies a different cyclic BB method and therefore does not resolve the CSD assertion.

Thus the established progress is general convergence for delayed/cyclic gradient frameworks and a two-dimensional cycle-boundary superlinear subsequence result. The remaining gap is the exact all-dimensional R-superlinear convergence assertion with the threshold m >= (n+1)/2; no still-open narrower subproblem was explicitly identified in a citable source with sufficient confidence for separate output files.

References
  1. [1]ICM Conjectures · 2022-opt-18 link
  2. [2]ICM 2022: An overview of nonlinear optimization link

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