Overview
Golden section search keeps interior points spaced by the golden ratio so one function evaluation can be reused each iteration.
CoreSource checked
Golden Section Search
Finds the minimum of a unimodal one-dimensional function by shrinking an interval efficiently.
Quick Facts
Compact metadata for implementation and discovery.
- Category
- Optimization
- Difficulty
- Beginner
- Family
- One-Dimensional Search
- Tier
- Core
- Status
- Verified
- Review
- Source Checked
- Sources
- 1
- Formulas
- 1
- Relationships
- 3
- Code example
- Ready
- Visualization
- Ready
Core Concepts
unimodal functioninterval reductiongolden ratiofunction reuse
Quick facts
Complexity
Runtime and memory behavior at a glance.
Best
O(log((b-a)/tolerance))
Average
O(log((b-a)/tolerance))
Worst
O(log((b-a)/tolerance))
Space
O(1)
When to Use
- Use when tuning model parameters needs unimodal function.
- Use when the input matches the assumptions behind one-Dimensional Search.
When Not to Use
- Avoid when the objective is cheap and convex enough for a simpler exact solver.
- Avoid when a simpler algorithm gives the required result with less setup.
What It Does
- Finds the minimum of a unimodal one-dimensional function by shrinking an interval efficiently
- Uses unimodal function and interval reduction to guide the work.
- Produces best candidate.
Use Cases
- Tuning model parameters
- Searching design spaces
- Fitting models to noisy data
Inputs
- Objective function
- Initial candidate or population
- Stopping rule
Outputs
- Best candidate
- Objective value
- Search history
Key Formulas (1)
Golden ratio split
c = b − (b − a) / φ
c = b - (b - a) / phiPlaces an interior point so shrinking the interval reuses one evaluation.
Pseudocode
- 1.Prepare the inputs and state needed for Golden Section Search.
- 2.Apply the unimodal function step while the stopping condition has not been met.
- 3.Update intermediate state and discard invalid candidates.
- 4.Return best candidate.
Canonical Example
- Input
- Golden Section Search input with a small representative case
- Output
- Best candidate
- Explanation
- Golden Section Search applies unimodal function to produce best candidate.
Implementation Notes
- Track both best-so-far and current candidate state.
- Keep the unimodal function step separate enough to test independently.
Common Pitfalls
- Stopping too early can make a poor local result look like convergence.
- Forgetting to check edge cases can make Golden Section Search look correct on only ideal inputs.
Interactive Mechanism Walkthrough
Iterative Convergence & State TrajectoryPlot candidate points, objective values, and improvement steps across iterations.
Step 1 of 4: Prepare the inputs and state needed for Golden Section Search.
optimization-mechanismNotes
Golden Section Search should be used with the assumptions and tradeoffs of one-Dimensional Search in mind.
Sources (1)
What each reference supports for this entry.
Numerical Recipes
BookNumerical methods reference for scalar optimization, interpolation, integration, and practical algorithm tradeoffs.
Open sourceExplore Next
Good next jumps from the relationship graph, ranked for useful discovery.
Relationships
View full relationship pageAlternative toBisection Method
Golden-section search optimizes a unimodal function, while bisection finds a bracketed root.
Often paired withPowell's Method
Derivative-free line searches can be used inside direction-set methods.
Related Algorithms
Tags
OptimizationSearchRoot FindingIterative