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.

Version: v2 · 2026-09-19 · History
Domain-specific #: 4749
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Numerical Optimization, Univariate Search → Mathematics

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.

How would you explain it like I'm…

Hilltop Hunting Game

Imagine looking for the very top of a single hill along a path, but you can only check heights at a few spots. You check two spots, and the lower one tells you which end of the path can't hold the top, so you throw that end away. If you pick your spots with a special spacing, one old check can be reused next time, so you only need one new check each round.

Golden Ratio Peak Finder

Golden-section search is a way to find the highest (or lowest) point of a curve that has just one peak (or valley) in a stretch. You check two points inside the stretch and compare them. The side beyond the lower point can't hold the peak, so you cut it off. The two points are spaced using a special number called the golden ratio, so the leftover stretch looks like a smaller copy of the old one and one of your old checks lands exactly where a new check is needed. That means each step costs just one new measurement, and the stretch shrinks by the same fraction every time.

Self-Similar Bracketing Search

Golden-section search is a method for finding the extremum of a one-variable function on an interval, assuming the function is unimodal there, meaning it has a single peak or valley. It keeps a bracket that must contain the optimum, evaluates two interior points, and discards the side that unimodality rules out. The points are spaced by the golden ratio, so the remaining interval is geometrically similar to the previous one, shrinking by the same fixed factor each step. This similarity lets one old point become an interior point of the new interval, so only one new function evaluation is needed per step. Unlike methods such as Newton's, it needs no derivatives. If the function isn't unimodal, the bracket can discard the true optimum.

Golden-section search is a derivative-free bracketing method for locating an extremum of a one-dimensional function on an interval where it is unimodal. It maintains a bracket [a, b] and two interior points placed at golden-ratio fractions of the interval. Comparing function values at those points identifies a subinterval that, by unimodality, cannot contain the optimum, and it is discarded. The golden-ratio placement makes the retained interval geometrically similar to the previous one, so one previously evaluated interior point becomes an interior point of the new bracket. Hence each iteration requires only one new function evaluation, and the bracket length contracts by a fixed ratio. Its defining features are thus preserved bracketing under the unimodality assumption, constant-ratio contraction and evaluation reuse via self-similar placement. Without unimodality, the discard step is not justified and the method can lose the optimum.

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.
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) — outgoing typed relations

Golden-section search is a kind of Optimization Prime

The algorithm searches for an extremum under a bracket.

Hierarchy path (1) — routes to 1 parentless root

Golden-section search → is a kind of Optimization

Neighborhood in Abstraction Space¶

Golden-section search sits in a crowded region of the domain-specific corpus (32^nd percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Optimization Algorithms & Methods (19 abstractions)

Nearest neighbors

Ternary search — 0.88
Piecewise function — 0.88
Total variation — 0.88
Interchange of limiting operations — 0.88
Singular function — 0.88

Computed from structural-signature embeddings · 2026-10-25