DIRECT global optimization

⚠️ Curated Limitations & Negative Results

• This record describes the original 1993 DIRECT construction. Later locally biased, tie-reduced, restart, noisy-objective, constrained or parallel variants are not silently imported or declared exact aliases. A direct-search optimizer is not the maintained DirectPreferenceOptimization algorithm.
• The source uses finite-valued deterministic objective evaluations on a finite box. General constraints, failed evaluations, noise handling, categorical parameters, zero-width coordinates and floating-point degeneracy need additional implementation rules. Normalization and tie conventions can affect the search trajectory; no rotation invariance or coordinate-independent behavior is asserted.
• The lower-right convex-hull interpretation concerns existence of a positive weight satisfying both inequalities. A point touching only at weight zero is excluded by the stated definition, and the epsilon filter can exclude a hull candidate. Equal-distance candidates require direct value comparison rather than division by a zero distance difference.
• In the introductory Shubert comparison, PDF p.4 / printed p.160 Eq 5 omits the factor one half on K(b-a) for the claimed bottom-of-V value. With equal endpoint values 0, interval [0,2] and K=1, the intersecting lines have minimum -1, while the printed expression gives -2. That expression is a weaker lower bound in this example, not the stated intersection value. DIRECT’s center-based Eq 8 and potentially optimal definitions are separate.
• The convergence proof’s Eq 11, PDF p.17 / printed p.173, writes an unrestricted maximum of ratios with denominator d_j-d_i, including equal-distance cases and i=j. The argument requires taking those ratios only over strictly smaller distances, while handling equal-distance centers by selecting a minimum-value one. This is an explicit curator reading of the definition-level proof, not an asserted author-issued correction.
• Dense sampling alone does not guarantee convergence of values for arbitrary discontinuities: a function equal to -1 only at sqrt(2)/2 and 0 elsewhere on [0,1] has an isolated minimum that the exact rational center sequence never samples. Even for a continuous objective, a finite set of sampled values can miss a narrow negative tent between samples. Without additional certified function bounds, a budget stop cannot certify global accuracy.
• The source’s benchmark comparisons use different convergence definitions for competing methods and discuss parameter-tuning and stochastic-run comparability problems. No benchmark was reproduced and no universal speed, dimension-independent storage, parameter-insensitivity or superiority claim is made. The central sample avoids enumerating all box vertices, but subdivision, retained rectangles and coordinate bookkeeping still require resources.
• The original bounded calculator in research/discovery/direct-boundaries.py uses rational radius clouds for selection, exact multivariate partition geometry and finite one-dimensional examples. It does not implement general multivariate square-root radius selection or establish asymptotic results by testing. Its trace evaluation counts denote distinct sampled locations, not runtime function-call costs; center values may be recomputed for geometric checks. It is supporting evidence within the author workflow, not independent scientific approval or external author-code execution.