text: Theorem 6 proves a constructive Omega(1/sqrt(epsilon)) lower bound on the number of iterations for Frank-Wolfe with short steps or exact line search when both the objective and the constraint set are smooth and strongly convex (Theorem 6).

text: The lower-bound construction uses the specific instance of minimizing f(x) = ||x-p||_2^2 over the Euclidean unit ball B_1(0) with the unconstrained minimizer p on the boundary (||p||=1), showing this hard instance matches the known O(1/sqrt(epsilon)) upper bound of Garber et al. (2015) (Theorem 6).

text: The lower bound is established via a 'backward reconstruction' technique that builds worst-case trajectories by working backward from terminal iterates, producing precise sequences over horizons of 10,000+ iterations (Section on Backward Reconstruction Strategy).

text: The analysis reduces the dynamics to a two-dimensional invariant subspace parametrized by residual magnitude r and contraction factor s, revealing stable phases of monotonically increasing contraction followed by intermittent 'jumps' that explain the convergence stalling behavior (Section on Dynamical Analysis).

text: The result shows strong convexity of the constraint set alone cannot yield convergence rates uniformly faster than O(1/sqrt(epsilon)), and this conclusion holds dimension-independently and extends to ellipsoidal constraints via affine invariance (Theorem 6, Implications).

scope: Finite-precision execution corroborates the constructive witness; the theorem remains an analytical result.

status: numerically corroborated

scope: The same hard instance is bracketed by matching residual envelopes and has a terminal-gap slope near -2.

scope: The largest generated trajectory exceeds 10,000 transitions and every adjacent pair is forward/backward checked.

status: reproduced

scope: The lower-bound witness is jump-free; intermittent jumps are demonstrated on a separate feasible trajectory and are not conflated with the witness.

status: reproduced with qualification

scope: Numerical embeddings and three ellipsoid maps corroborate the theorem's dimension and affine consequences.

honesty_note: The backward lower-bound witness is jump-free. The jump table is a separate feasible forward trajectory illustrating the paper's intermittent-jump regime; it is not represented as part of the lower-bound witness.

method: independent Python/mpmath translation of displayed paper equations

paper_title: Lower Bounds for Frank-Wolfe on Strongly Convex Sets


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method: independent Python/mpmath translation; author Julia source not executed