Golden-section search¶
A derivative-free interval-reduction algorithm for optimizing a unimodal function by placing interior evaluations in the golden ratio so one point can be reused each iteration.
Core Idea¶
Golden-section search locates a one-dimensional extremum by repeatedly discarding the subinterval inconsistent with unimodality using golden-ratio-spaced samples. The golden ratio makes the retained interval geometrically similar to the previous one, so one old function value becomes an interior value in the next step and only one new evaluation is needed. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.
Scope of Application¶
Golden-section search belongs to numerical optimization and is useful where the analyst can specify a real-valued function, closed bracket containing a unimodal optimum, two interior test points, function evaluations, golden ratio, interval update rule and stopping tolerance, then evaluate the optimum remains bracketed under the unimodality assumption and interval length contracts by a fixed ratio. The scope is broad within that domain but bounded by the need for the optimum remains bracketed under the unimodality assumption and interval length contracts by a fixed ratio. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
Clarity¶
The abstraction clarifies a crowded vocabulary by making the optimum remains bracketed under the unimodality assumption and interval length contracts by a fixed ratio the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Golden-section search can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.
Manages Complexity¶
Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Golden-section search. Golden-section search compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.
Abstract Reasoning¶
- Identify the carrier. State what the elements, states, objects, or observations are: a real-valued function, closed bracket containing a unimodal optimum, two interior test points, function evaluations, golden ratio, interval update rule and stopping tolerance. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the optimum remains bracketed under the unimodality assumption and interval length contracts by a fixed ratio independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of numerical optimization because they reuse a real-valued function, closed bracket containing a unimodal optimum, two interior test points, function evaluations, golden ratio, interval update rule and stopping tolerance, The golden ratio makes the retained interval geometrically similar to the previous one, so one old function value becomes an interior value in the next step and only one new evaluation is needed., and type the carrier, state every parameter and convention in the definition, test that the optimum remains bracketed under the unimodality assumption and interval length contracts by a fixed ratio, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Golden-section search Domain-specific
Parents (1) — more general patterns this builds on
-
Golden-section search is a kind of Optimization Prime
The proposed strict upward parent is
prime:optimization.
Hierarchy path (1) — routes to 1 parentless root
- Golden-section search → Optimization
Neighborhood in Abstraction Space¶
Golden-section search sits in a moderately populated region (51st percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Nonlinear & Simulation Optimization (7 abstractions)
Nearest neighbors
- Ternary search — 0.88
- Piecewise function — 0.88
- Total variation — 0.88
- Interchange of limiting operations — 0.88
- Continuous function (ordinal theory) — 0.88
Computed from structural-signature embeddings · 2026-09-08