A Simplex Method for Function Minimization

1965/01/01 by J. A. Nelder, R. Mead · 28,959 citations
Engineering · Mathematics · Physics and Astronomy · #Algorithm #Applied mathematics #Combinatorics #Computer science #Control Systems and Identification #Function (biology) #Graph #Hessian matrix #Linear programming #Mathematical analysis #Mathematical optimization #Mathematics #Minification #Neighbourhood (mathematics) #Scientific Research and Discoveries #Simplex #Simplex algorithm #Statistical and numerical algorithms #Vertex (graph theory)

paper · doi:10.1093/comjnl/7.4.308

published in The Computer Journal 7(4), 308-313 (Oxford University Press)

openalex publication_date 1965/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06

Abstract

A method is described for the minimization of a function of n variables, which depends on the comparison of function values at the (n + 1) vertices of a general simplex, followed by the replacement of the vertex with the highest value by another point. The simplex adapts itself to the local landscape, and contracts on to the final minimum. The method is shown to be effective and computationally compact. A procedure is given for the estimation of the Hessian matrix in the neighbourhood of the minimum, needed in statistical estimation problems.

Citations

• An Automatic Method for Finding the Greatest or Least Value of a Function

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• Bayesian Forensic DNA Mixture Deconvolution Using a Novel String Similarity Measure
• Patient positioning with X-ray detector self-calibration for image guided therapy
• New approximation for GARCH parameters estimate
• Bayeswave: Bayesian inference for gravitational wave bursts and instrument glitches
• Embedding the Zee-Wolfenstein neutrino mass matrix in anSO(10)×A4GUT scenario
• Multisurface Adiabatic Reactive Molecular Dynamics
• Instanton analysis of Low-Density-Parity-Check codes in the error-floor regime
• Sequential Recruitment and Combinatorial Assembling of Multiprotein Complexes in Transcriptional Activation
• Fracture in Westerly Granite under AE Feedback and Constant Strain Rate Loading: Nucleation, Quasi-static Propagation, and the Transition to Unstable Fracture Propagation
• Integrating Atlas and Graph Cut Methods for LV Segmentation from Cardiac Cine MRI
• Mixtures of bosonic and fermionic atoms in optical lattices
• A Deep Multicolor Survey. V. The M Dwarf Luminosity Function
• Discovery of a Physically Interpretable Data-Driven Wind-Turbine Wake Model
• APS 9: an improved adaptive population-based simplex method for real-world engineering optimization problems
• Predicting the unpredictable: Interpreting neutralized segments in Dutch
• Efficient Statistical Inference for Stochastic Reaction Processes
• A4 model for the quark mass matrices
• A Posterior Predictive Approach to Multiple Response Surface Optimization
• VFISV Inversion Code Documentation for SOLIS/VSM Pipeline Implementation
• Stationary probability density of stochastic search processes in global optimization
• Hybrid algorithm based optimization strategies for analog circuit sizing in low dropout regulators
• Estimating Cosmological Parameters from the Dark Matter Distribution
• Software package MaRDI Open Interfaces for improved interoperability in numerical optimization
• MaRDI Open Interfaces for Interoperable Nonlinear Optimization
• A general cost-benefit-based adaptation framework for multimeme algorithms
• THE CENTURY SURVEY GALACTIC HALO PROJECT. III. A COMPLETE 4300 DEG2SURVEY OF BLUE HORIZONTAL BRANCH STARS IN THE METAL-WEAK THICK DISK AND INNER HALO
• The ACS Virgo Cluster Survey. VI. Isophotal Analysis and the Structure of Early‐Type Galaxies
• THE SEGUE STELLAR PARAMETER PIPELINE. I. DESCRIPTION AND COMPARISON OF INDIVIDUAL METHODS
• Experimental constraints on the generation of FeTi basalts, andesites, and rhyodacites at the Galapagos Spreading Center, 85°W and 95°W
• Bell inequalities resistant to detector inefficiency
• Shadow Imaging of Transiting Objects
• Detection of Period Variations in Extrasolar Transiting Planet OGLE-TR-111b
• A Transiting Planet of a Sun‐like Star
• Correlations of the elements of the neutrino mass matrix
• Variations of NO, NO2 and O3 concentrations downwind of a Los Angeles freeway
• Structure and kinematics of edge-on galaxy discs - I. Observations of the stellar kinematics
• Plasma and salivary pharmacokinetics of caffeine in man
• The absolute bioavailability of caffeine in man
• Neuro-Symbolic ODE Discovery with Latent Grammar Flow
• Interaction of solitons in one-dimensional dipolar Bose-Einstein condensates and formation of soliton molecules
• Proactive interference in aging: A model-based study
• Predictive performance of viscous potential functions for modeling strain rate sensitivity of soft materials
• Stellar populations in the surrounding field of the LMC clusters NGC 2154 and NGC 1898
• On the Design and Implementation of Higher Order Differential Microphones
• A diffusion model analysis of adult age differences in episodic and semantic long-term memory retrieval.
• The mutual orbit, mass, and density of transneptunian binary Gǃkúnǁ'hòmdímà (229762 2007 UK126)
• Physical properties ofβLyrae A and its opaque accretion disk
• BBOWP-Bench: Evaluating LLMs on Black-Box Optimization Word Problems
• Tailoring Absorption in Metal Gratings with Resonant Ultrathin Bridges
• ldmppr: Location Dependent Marked Point Processes in R
• Iterative Sparse Asymptotic Minimum Variance Based Approaches for Array Processing
• Simultaneous extraction of GaAs MESFET channel and gate diode parameters and its application to circuit simulation
• Traversable Wormhole Geometry Reconstruction from the Rotation Curve of NGC 3198: A Comparative Study of Dark Matter Halo Profiles
• Seven temperate terrestrial planets around the nearby ultracool dwarf star TRAPPIST-1
• Regular Radial Velocity Variations in Nine G- and K-type Giant Stars: Eight Planets and One Planet Candidate
• SAS-Prompt: Large Language Models as Numerical Optimizers for Robot Self-Improvement
• How Forward‐Scattering Snow and Terrain Change the Alpine Radiation Balance With Application to Solar Panels
• Warm terrestrial planet with half the mass of Venus transiting a nearby star
• A Review of Variational Quantum Algorithms: Insights into Fault-Tolerant Quantum Computing
• p-mode frequencies in solar-like stars
• Fisher Scoring for Exact Matérn Covariance Estimation through Stable Smoothness Optimization
• A Conversation with John Nelder
• A Stochastic Differential Equation Model for the Height Growth of Forest Stands
• A Biologically Meaningful Approach to Response Surface Analysis
• Keeping it Cool: Much Orbit Migration, yet Little Heating, in the Galactic Disk
• Mechanistic Elements of Protein Cold Denaturation
• Quantum Circuit Learning Using Non-Integrable System Dynamics
• A Survey on Policy Search for Robotics
• Spectral sensitivities of photoreceptors and their role in colour discrimination in the green-backed firecrown hummingbird (Sephanoides sephaniodes)
• Continental-scale mapping of Adélie penguin colonies from Landsat imagery
• Crystal lattice properties fully determine short-range interaction parameters for alkali and halide ions
• Spatial modeling of bone microarchitecture
• MECI: A Method for Eclipsing Component Identification
• MODIFICATION OF A CHARGED-BOSE-GAS MODEL FOR OBSERVED ROOM-TEMPERATURE SUPERCONDUCTIVITY IN NARROW CHANNELS THROUGH FILMS OF OXIDIZED ATACTIC POLYPROPYLENE
• Spectroscopic atlas of Hαand Hβin a sample of northern Be stars
• Recovering the star formation rate in the solar neighborhood
• Multilayer perceptrons and radial basis function neural network methods for the solution of differential equations: A survey
• The Developmental Dynamics of Terrorist Organizations
• ULTRALUMINOUS X-RAY SOURCES IN ARP 147
• Modelling the Spread in Space and Time of An Airborne Plant Disease
• Latent Variable Phillips Curve
• Calibration of a Macroscopic Coupled People-Epidemic Transport PDE Model via Density and Velocity Computation from Microscopic Data
• On the Folded Normal Distribution
• Using models to improve optimizers for variational quantum algorithms
• Cross-dialectal data sharing for acoustic modeling in Arabic speech recognition
• Surface composition and properties of Ganymede: Updates from ground-based observations with the near-infrared imaging spectrometer SINFONI/VLT/ESO
• Global Likelihood Optimization Via the Cross-Entropy Method, with an Application to Mixture Models
• The Star Formation History of the Local Group Dwarf Galaxy Leo I
• Heuristic for estimation of multiqubit genuine multipartite entanglement
• NORMAL GLOBULAR CLUSTER SYSTEMS IN MASSIVE LOW SURFACE BRIGHTNESS GALAXIES
• Model-Based Clustering of Multivariate Rating Data Accounting for Feeling and Uncertainty
• Accelerated Degradation Testing With Long-Term Memory Effects
• Kondo Breakdown in a Spin-1/2 Chain of Adatoms on a Dirac Semimetal
• PHast -- Perfect Hashing made fast
• Model Choice Matters for Age Inference on the Red Giant Branch
• CosmoSIS: Modular cosmological parameter estimation
• Categorical Data Analysis Using a Skewed Weibull Regression Model
• Banyan. X. Discovery of a Wide, Low-gravity L-type Companion to a Fast-rotating M3 Dwarf*
• TRANSIT MODEL OF PLANETS WITH MOON AND RING SYSTEMS
• maxsmooth: rapid maximally smooth function fitting with applications in Global 21-cm cosmology
• Einstein–Cartan, Bianchi I and the Hubble diagram
• The cosmological analysis of X-ray cluster surveys
• The McDonald Accelerating Stars Survey (MASS): White Dwarf Companions Accelerating the Sun-like Stars 12 Psc and HD 159062
• Variational calculation of the ground state of closed-shell nuclei up to A=40
• Estimating the parameters of nonspinning binary black holes using ground-based gravitational-wave detectors: Statistical errors
• A non-iterative method for optimizing linear functions in convex linearly constrained spaces
• Force correcting atom centred potentials for generalised gradient approximated density functional theory: Approaching hybrid functional accuracy for geometries and harmonic frequencies in small chlorofluorocarbons
• Constraining the surface properties of Helene
• Flexible Estimation of the Heterogeneous Non-Parametric Component in a Relative Survival Cure Model
• Koopman-Based Nonlinear Identification and Model Predictive Control of a Turbofan Engine
• Preconditioning Natural and Second Order Gradient Descent in Quantum Optimization: A Performance Benchmark
• Learning Underwater Active Perception in Simulation
• Efficient Design of Compliant Mechanisms Using Multi-Objective Optimization
• Optimized Rhombic Experimental Dynamic Checkerboard Designs to Elucidate Pharmacodynamic Drug Interactions of Antibiotics
• A variational eigenvalue solver on a photonic quantum processor
• Differential Flatness of Quadrotor Dynamics Subject to Rotor Drag for Accurate Tracking of High-Speed Trajectories
• Modeling Stochastic Variability in Multiband Time-series Data
• Deep phase modulation interferometry
• Multilevel Artificial Neural Network Training for Spatially Correlated Learning
• Numerical Derivatives, Projection Coefficients, and Truncation Errors in Analytic Hilbert Space With Gaussian Measure
• GRADIENT-BASED STOCHASTIC OPTIMIZATION METHODS IN BAYESIAN EXPERIMENTAL DESIGN
• Best-Practice Aspects of Quantum-Computer Calculations: A Case Study of the Hydrogen Molecule
• Detection of titanium oxide in the atmosphere of a hot Jupiter
• Benchmarking NLopt and state-of-the-art algorithms for continuous global optimization viaIACOR
• Optimisation of the event-based TOF filtered back-projection for online imaging in total-body J-PET
• SARAS 2: a spectral radiometer for probing cosmic dawn and the epoch of reionization through detection of the global 21-cm signal
• Adaptive Circuit Learning for Quantum Metrology
• A systematic approach to Kähler moduli stabilisation
• Architectural Techniques for Improving NAND Flash Memory Reliability
• Spectroscopic analysis of southern B and Be stars
• Monte Carlo integration with adaptive variance selection for improved stochastic efficient global optimization
• Quantification of Model Uncertainty on Path-Space via Goal-Oriented\n Relative Entropy
• Quantum Approximate Optimization Algorithm: Performance, Mechanism, and Implementation on Near-Term Devices
• Characterization of the stochastic signal originating from compact binary populations as measured by LISA
• Typical and extreme entropies of long-lived isolated quantum systems
• Coarse-to-Fine Registration of Airborne LiDAR Data and Optical Imagery on Urban Scenes
• Towards accurate modelling of galaxy clustering on small scales: testing the standard ΛCDM + halo model
• The CARMENES search for exoplanets around M dwarfs
• Combinatorial Image Analysis
• A mutagenetic tree hidden Markov model for longitudinal clonal HIV sequence data
• Interplanetary particle transport simulation for warning system for aviation exposure to solar energetic particles
• Optimal Decision-Making in Mixed-Agent Partially Observable Stochastic Environments via Reinforcement Learning
• Halos of Spiral Galaxies. I. The Tip of the Red Giant Branch as a Distance Indicator
• Quantum optimal control with quantum computers: A hybrid algorithm featuring machine learning optimization
• Defect-mediated emulsification in two dimensions
• Asymmetric broadcasting of quantum correlations
• RADIO ASTROMETRY OF THE TRIPLE SYSTEMS ALGOL AND UX ARIETIS
• The luminosity-dependent high-redshift turnover in the steep spectrum radio luminosity function: clear evidence for downsizing in the radio-AGN population
• Analysis techniques for complex field radiation pattern measurements
• Comparative study of quantum discord and geometric discord for generic bipartite states
• Longitudinal quantile regression in presence of informative drop-out\n through longitudinal-survival joint modeling
• Characterizing stellar halo populations II: the age gradient in blue horizontal-branch stars
• Extending the dynamic range of SiPMs by understanding their non-linear behavior
• Quantum variational optimization: The role of entanglement and problem hardness
• Magnetism and exchange interaction of small rare-earth clusters; Tb as a representative
• Beach-level 24-hour forecasts of Florida red tide-induced respiratory irritation
• Gaia-ESO Survey: Analysis of pre-main sequence stellar spectra
• Optimization of a Controlled-Z Gate with Data-Driven Gradient-Ascent Pulse Engineering in a Superconducting-Qubit System
• Internal rotation of 13 low-mass low-luminosity red giants in theKeplerfield
• Updated global 3+1 analysis of short-baseline neutrino oscillations
• Easy Hyperparameter Search Using Optunity
• Deep Scatter Estimation (DSE): Accurate Real-Time Scatter Estimation for X-Ray CT Using a Deep Convolutional Neural Network
• Comparison of a modified vibro-impact nonlinear energy sink with other kinds of NESs
• COCO: a platform for comparing continuous optimizers in a black-box setting
• Bimodal radio variability in OVRO-40 m-monitored blazars
• Optimizing quantum codes with an application to the loss channel with partial erasure information
• The orbit of transneptunian binary Manwë and Thorondor and their upcoming mutual events
• Semiparametric Two-Component Mixture Models When One Component Is Defined Through Linear Constraints
• A Meta-learning Approach to Reservoir Computing: Time Series Prediction with Limited Data
• DEEP MMT TRANSIT SURVEY OF THE OPEN CLUSTER M37. III. STELLAR ROTATION AT 550 Myr
• A composite parameterization of unitary groups, density matrices and subspaces
• THE HOT INTERSTELLAR MEDIUM OF THE INTERACTING GALAXY NGC 4490
• Upgrading from Gaussian Processes to Student’s-T Processes
• Design and Performance Analysis of a Highly Efficient Polychromatic Full Stokes Polarization Modulator for the CRISP Imaging Spectrometer
• Extracting the Subhalo Mass Function from Strong Lens Images with Image Segmentation
• Particle acceleration in low-power hotspots: modelling the broad-band spectral energy distribution
• Analytic semi-device-independent entanglement quantification for bipartite quantum states
• Measurement-based cooling of a nonlinear mechanical resonator
• Thermal geometric discords in a two-qutrit system
• Structures and Topological Transitions of Hydrocarbon Films on Quasicrystalline Surfaces
• Violations of Local Realism by Two EntangledN-Dimensional Systems Are Stronger than for Two Qubits
• Global-scale phylogenetic linguistic inference from lexical resources
• Ultrafast critical ground state preparation via bang–bang protocols
• The Lowest Mass White Dwarf
• Multi-scale perimeter control approach in a connected-vehicle environment
• Learning, Computing, and Trustworthiness in Intelligent IoT Environments: Performance-Energy Tradeoffs
• Experimental Verification of Electron-Photon Entanglement
• Extension of openCOSMO-RS into a full open-source equation of state: Implementation, parameterization, and benchmarking
• A spatial Poisson hurdle model with application to wildfires
• Iterative Hybrid Discrete-Continuous Viewpoint Planning for UAV Photogrammetry
• Parameters Fixing Strategy for Quantum Approximate Optimization Algorithm
• Deep extragalactic visible legacy survey (DEVILS): stellar mass growth by morphological type since z = 1
• Bregman Linearized Augmented Lagrangian Method for Nonconvex Constrained Stochastic Zeroth-order Optimization
• Mutual orbits and masses of six transneptunian binaries
• Optimum design of permeable diffractive lenses based on photon sieves
• Structural and electronic properties of small neutral(MgO)nclusters
• Structure and bonding in small neutral alkali halide clusters
• PINNverse: Accurate parameter estimation in differential equations from noisy data with constrained physics-informed neural networks
• Static and dynamic error of a biplanar videoradiography system using marker-based and markerless tracking techniques. [europepmc]
• Flexing computational muscle: modeling and simulation of musculotendon dynamics. [europepmc]
• T1 and extracellular volume mapping in the heart: estimation of error maps and the influence of noise on precision. [europepmc]
• Larger extrastriate population receptive fields in autism spectrum disorders. [europepmc]
• Cross-validation pitfalls when selecting and assessing regression and classification models. [europepmc]
• The preclinical Alzheimer cognitive composite: measuring amyloid-related decline. [europepmc]
• Uncertainty in perception and the Hierarchical Gaussian Filter. [europepmc]
• Assessing cognitive processes with diffusion model analyses: a tutorial based on fast-dm-30. [europepmc]
• Emergence and potential for spread of Chikungunya virus in Brazil. [europepmc]
• Dynamical feature extraction at the sensory periphery guides chemotaxis. [europepmc]
• Methods for Health Economic Evaluation of Vaccines and Immunization Decision Frameworks: A Consensus Framework from a European Vaccine Economics Community. [europepmc]
• CHiCAGO: robust detection of DNA looping interactions in Capture Hi-C data. [europepmc]
• Learning Reward Uncertainty in the Basal Ganglia. [europepmc]
• Mobile Sensing and Support for People With Depression: A Pilot Trial in the Wild. [europepmc]
• Genome-scale rates of evolutionary change in bacteria. [europepmc]
• Testing the validity of conflict drift-diffusion models for use in estimating cognitive processes: A parameter-recovery study. [europepmc]
• Advances in the Development of Shape Similarity Methods and Their Application in Drug Discovery. [europepmc]
• New tools for automated high-resolution cryo-EM structure determination in RELION-3. [europepmc]
• Inertial delay of self-propelled particles. [europepmc]
• Estimating Relatedness Between Malaria Parasites. [europepmc]
• Ancestry-agnostic estimation of DNA sample contamination from sequence reads. [europepmc]
• A flexible framework for simulating and fitting generalized drift-diffusion models. [europepmc]
• Bifurcation in brain dynamics reveals a signature of conscious processing independent of report. [europepmc]
• Cost function dependent barren plateaus in shallow parametrized quantum circuits. [europepmc]
• Self-Driving Laboratories for Chemistry and Materials Science. [europepmc]

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