Newton Optimisation Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Newton Optimisation Ontology within the Kireth Research Semantic Library. Newton optimisation is a second-order numerical optimisation framework that uses gradient and Hessian information to construct a local quadratic model and compute a search step from a linear system. This ontology supports unconstrained and constrained relationships, exact and modified Newton methods, damped Newton, line search, trust-region variants, Hessian regularisation, indefinite Hessians, sparse and matrix-free implementations, local quadratic convergence, conditioning, stopping criteria, verification and reproducibility.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Newton Optimisation Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Computational Mathematics"
 SPECIALISATION = "Optimisation Computation / Newton Optimisation"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/computational-mathematics-ontologies/optimisation-computation/newton-optimisation-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = NewtonOptimisation
 ENTITY_CLASS = SecondOrderOptimisationFramework
 PRIMARY_CONTEXT = ComputationalOptimisation
 DEFINITION = "Newton optimisation is a second-order iterative method that computes optimisation steps from the gradient and Hessian of an objective or Lagrangian."
 SYMBOLIC_FORMS = [
  "H(x_k)p_k=-grad f(x_k)",
  "x_(k+1)=x_k+p_k",
  "m_k(p)=f(x_k)+g_k^T p+0.5*p^T H_k p",
  "grad f(x_star)=0"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for second-order local models, Hessian systems, damped and modified Newton steps, line-search and trust-region globalisation, conditioning, local convergence and verification."
}

SCOPE {
 INCLUDE = [
  NewtonOptimisation,NewtonMethodForOptimisation,SecondOrderMethod,
  UnconstrainedOptimisation,ConstrainedOptimisation,
  ObjectiveFunction,DecisionVariable,Gradient,Hessian,
  HessianMatrix,GradientVector,SearchDirection,NewtonDirection,
  NewtonStep,FullNewtonStep,DampedNewtonStep,
  ModifiedNewtonMethod,RegularisedNewtonMethod,
  InexactNewtonMethod,TruncatedNewtonMethod,
  NewtonCG,NewtonKrylovOptimisation,
  LineSearchNewton,DampedNewton,BacktrackingLineSearch,
  ArmijoCondition,WolfeCondition,StrongWolfeCondition,
  TrustRegionNewton,TrustRegion,TrustRegionRadius,
  TrustRegionSubproblem,CauchyPoint,DoglegMethod,
  SteihaugMethod,TruncatedConjugateGradient,
  LocalQuadraticModel,TaylorExpansion,SecondOrderTaylorModel,
  PositiveDefiniteHessian,PositiveSemidefiniteHessian,
  IndefiniteHessian,NegativeCurvature,HessianModification,
  ModifiedCholesky,DiagonalShift,LevenbergRegularisation,
  DampingParameter,RegularisationParameter,
  StationaryPoint,CriticalPoint,LocalMinimum,LocalMaximum,
  SaddlePoint,SecondOrderCriticalPoint,
  FirstOrderOptimality,SecondOrderOptimality,
  StrongConvexity,ConvexFunction,NonconvexFunction,
  LocalConvergence,QuadraticConvergence,SuperlinearConvergence,
  LinearConvergence,ConvergenceRegion,BasinOfAttraction,
  InitialGuess,Initialisation,StepAcceptance,
  PredictedReduction,ActualReduction,ReductionRatio,
  LinearSystem,LinearSolver,DirectSolver,IterativeSolver,
  CholeskyFactorisation,LDLFactorisation,QRFactorisation,
  ConjugateGradientMethod,MINRES,GMRES,
  SparseHessian,SparseMatrix,SparsityPattern,
  MatrixFreeMethod,HessianVectorProduct,
  AutomaticDifferentiation,NumericalDifferentiation,
  FiniteDifferenceGradient,FiniteDifferenceHessian,
  ExactHessian,ApproximateHessian,
  Scaling,VariableScaling,ObjectiveScaling,
  Conditioning,ConditionNumber,IllConditioning,
  Preconditioner,Preconditioning,
  GradientNorm,StepNorm,ObjectiveChange,
  NewtonDecrement,OptimalityResidual,
  StoppingCriterion,GradientTolerance,StepTolerance,
  FunctionTolerance,MaximumIterations,MaximumTime,
  FunctionEvaluationCount,GradientEvaluationCount,
  HessianEvaluationCount,LinearSolveCount,
  Failure,Success,Warning,Stagnation,Divergence,
  FloatingPointArithmetic,MachinePrecision,RoundOffError,
  Overflow,Underflow,Verification,Benchmarking,
  ReferenceSolution,Reproducibility
 ]
 EXCLUDE_AS_PRIMARY = [
  NewtonRaphsonRootFindingWithoutOptimisationObjective,
  QuasiNewtonMethodWhenHessianApproximationIsPrimary,
  GradientDescentWithoutSecondOrderModel,
  GaussNewtonMethodWhenLeastSquaresStructureIsPrimary
 ]
 DISAMBIGUATION_RULE = "When an optimisation step is computed from a Hessian-based local quadratic model or the linear system H p = -g, interpret the context as Newton optimisation."
}

CORE_CONCEPTS {
 NewtonOptimisation = "Second-order optimisation using gradient and Hessian information."
 NewtonDirection = "Search direction solving the Newton linear system."
 Hessian = "Matrix of second derivatives defining local curvature."
 LocalQuadraticModel = "Second-order Taylor approximation used to model the objective near the current iterate."
 DampedNewton = "Newton method that scales or line-searches the full Newton direction."
 TrustRegionNewton = "Newton method restricting the step to a trust region."
 NegativeCurvature = "Direction indicating the Hessian is not positive semidefinite."
 QuadraticConvergence = "Local error contraction proportional to the square of the previous error under suitable assumptions."
}

RELATIONS {
 NewtonOptimisation --is_child_of--> ComputationalOptimisation
 NewtonOptimisation --is_type_of--> SecondOrderMethod
 NewtonOptimisation --uses--> Gradient
 NewtonOptimisation --uses--> Hessian
 Hessian --defines--> LocalQuadraticModel
 NewtonDirection --solves--> HessianLinearSystem
 LineSearchNewton --uses--> LineSearch
 TrustRegionNewton --uses--> TrustRegion
 IndefiniteHessian --can_produce--> NonDescentNewtonDirection
 HessianModification --addresses--> IndefiniteHessian
 ModifiedCholesky --produces--> PositiveDefiniteApproximation
 InexactNewtonMethod --uses--> ApproximateLinearSolve
 NewtonCG --uses--> ConjugateGradientMethod
 MatrixFreeMethod --uses--> HessianVectorProduct
 StrongConvexity --supports--> PositiveDefiniteHessianNearSolution
 Verification --checks--> NewtonOptimisationResult
}

OPERATORS {
 NEWTON_DIRECTION(g,H) := solve(H*p=-g)
 NEWTON_UPDATE(x,p) := x_next=x+p
 DAMPED_NEWTON_UPDATE(x,p,alpha) := x_next=x+alpha*p
 QUADRATIC_MODEL(f,g,H,p) := f+g^T*p+0.5*p^T*H*p
 TRUST_REGION_SUBPROBLEM(g,H,Delta) := min_p g^T*p+0.5*p^T*H*p subject_to ||p||<=Delta
 NEWTON_DECREMENT(g,H) := sqrt(g^T*H^(-1)*g)
 REDUCTION_RATIO(f_old,f_new,m0,mp) := (f_old-f_new)/(m0-mp)
 HESSIAN_SHIFT(H,lambda) := H_reg=H+lambda*I
}

AXIOMS {
 A1 = "Newton optimisation uses a local second-order model of the objective."
 A2 = "The classical Newton direction solves H p = -g."
 A3 = "A positive-definite Hessian makes the Newton direction a descent direction when the gradient is nonzero."
 A4 = "An indefinite Hessian can produce ascent or saddle-seeking directions."
 A5 = "Damping, line search or trust regions globalise Newton steps away from the local convergence region."
 A6 = "Near a nondegenerate local minimiser with suitable smoothness, exact Newton iterations can converge quadratically."
 A7 = "Inexact Newton methods trade linear-solve accuracy for computational efficiency."
 A8 = "Newton optimisation results require Hessian, conditioning, convergence and reproducibility validation."
}

REASONING {
 R1_CONTEXT = identify(objective,variables,smoothness,convexity,sparsity,scale,accuracy_goal)
 R2_DERIVATIVES = obtain(gradient_and_Hessian_from_analytic_AD_or_numerical_sources)
 R3_CURVATURE = assess(Hessian_positive_definiteness_indefiniteness_conditioning_and_negative_curvature)
 R4_LINEAR_SYSTEM = configure(direct_iterative_sparse_or_matrix_free_Hessian_solve)
 R5_DIRECTION = compute(Newton_or_modified_Newton_direction)
 R6_GLOBALISE = apply(full_step,line_search,damping,trust_region_or_regularisation)
 R7_ACCEPT = evaluate(actual_vs_predicted_reduction_and_step_conditions)
 R8_MONITOR = track(objective,gradient_norm,step_norm,Newton_decrement,conditioning_and_progress)
 R9_CONVERGENCE = classify(local_quadratic_superlinear_linear_stagnant_or_divergent_behaviour)
 R10_VALIDATE = check(second_order_conditions,reference_solution,sensitivity_and_reproducibility)
}

CONSTRAINTS {
 C1 = "The classical Newton step can fail when the Hessian is singular or nearly singular."
 C2 = "An indefinite Hessian can generate a non-descent direction."
 C3 = "Finite-difference Hessians can be inaccurate because of truncation and round-off error."
 C4 = "Poor scaling can severely degrade Hessian conditioning."
 C5 = "Full Newton steps can be unreliable far from a minimiser."
 C6 = "Loose inexact-solve tolerances can spoil local convergence."
 C7 = "Overly strict linear-solve tolerances can waste computation."
 C8 = "A small gradient norm alone does not prove local minimality in nonconvex problems."
 C9 = "Resource-limit termination is not equivalent to convergence."
 C10 = "Finite precision can limit attainable quadratic convergence near the numerical error floor."
}

SEMANTIC_EQUIVALENCE {
 "NewtonOptimisation" <=> "NewtonMethodForOptimisation"
 "DampedNewton" <=> "LineSearchNewtonInDampedContext"
 "NewtonCG" <=> "NewtonConjugateGradient"
 "TruncatedNewtonMethod" <=> "InexactNewtonKrylovMethod"
 "ModifiedNewtonMethod" <=> "RegularisedNewtonMethodInCurvatureCorrectionContext"
}

VALIDATION {
 V1 = "Objective function is explicit."
 V2 = "Gradient and Hessian sources are documented."
 V3 = "Initial guess is documented."
 V4 = "Linear solver and tolerances are documented."
 V5 = "Hessian definiteness or curvature handling is documented."
 V6 = "Line-search, damping or trust-region strategy is documented where used."
 V7 = "Gradient norm, step norm and objective progress are monitored."
 V8 = "Observed local convergence rate is checked where feasible."
 V9 = "Second-order conditions are assessed for nonconvex problems."
 V10 = "Results are reproducible under documented arithmetic and solver settings."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 NEWTON_OPTIMISATION_DEFINITION = 0.99
 HESSIAN_REASONING = 0.99
 LINEAR_SYSTEM_REASONING = 0.99
 LINE_SEARCH_REASONING = 0.99
 TRUST_REGION_REASONING = 0.99
 QUADRATIC_CONVERGENCE_REASONING = 0.99
 NONCONVEX_CURVATURE_REASONING = 0.98
 POLICY = "Reduce confidence when Hessian quality, definiteness, linear-solve tolerance, scaling, initialisation, globalisation or second-order validation are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,OntologyGrounding,NewtonOptimisationReasoning,
  HessianReasoning,NewtonDirectionReasoning,
  SecondOrderModelReasoning,LineSearchNewtonReasoning,
  DampedNewtonReasoning,TrustRegionNewtonReasoning,
  ModifiedNewtonReasoning,InexactNewtonReasoning,
  NewtonCGReasoning,NegativeCurvatureReasoning,
  QuadraticConvergenceReasoning,ConditioningReasoning,
  SparseHessianReasoning,MatrixFreeReasoning,
  StoppingCriterionReasoning,VerificationReasoning,
  KnowledgeGraphConstruction,RetrievalAugmentation,RecursiveInference
 ]
 NORMALISE {
  "Newton optimisation" -> NewtonOptimisation
  "Newton optimization" -> NewtonOptimisation
  "Newton method for optimisation" -> NewtonOptimisation
  "damped Newton" -> DampedNewton
  "Newton-CG" -> NewtonCG
  "truncated Newton" -> TruncatedNewtonMethod
 }
}

EXECUTION_RULES {
 E1 = "If an optimisation step is computed from Hessian and gradient information, activate NewtonOptimisationOntology."
 E2 = "Assess Hessian definiteness and conditioning before accepting the raw Newton direction."
 E3 = "If the Hessian is indefinite or singular, modify, regularise or trust-region the local model."
 E4 = "Use line search or trust-region control when far from the local convergence region."
 E5 = "For large sparse problems, exploit sparse factorisation, iterative solvers or Hessian-vector products."
 E6 = "For inexact Newton methods, coordinate linear-solve tolerance with nonlinear optimisation progress."
 E7 = "Monitor gradient norm, step norm, reduction ratio and conditioning."
 E8 = "Distinguish stationary points from verified local minima in nonconvex problems."
 E9 = "Validate second-order behaviour, local convergence and reproducibility."
}

RECURSION {
 ENTRY = NewtonOptimisation
 EXPAND(NewtonOptimisation) -> [
  NewtonMethodForOptimisation,SecondOrderMethod,ObjectiveFunction,
  DecisionVariable,Gradient,Hessian,HessianMatrix,SearchDirection,
  NewtonDirection,NewtonStep,FullNewtonStep,DampedNewtonStep,
  ModifiedNewtonMethod,RegularisedNewtonMethod,InexactNewtonMethod,
  TruncatedNewtonMethod,NewtonCG,NewtonKrylovOptimisation,
  LineSearchNewton,DampedNewton,BacktrackingLineSearch,
  ArmijoCondition,WolfeCondition,TrustRegionNewton,TrustRegion,
  TrustRegionRadius,TrustRegionSubproblem,CauchyPoint,DoglegMethod,
  SteihaugMethod,TruncatedConjugateGradient,LocalQuadraticModel,
  TaylorExpansion,PositiveDefiniteHessian,IndefiniteHessian,
  NegativeCurvature,HessianModification,ModifiedCholesky,
  DiagonalShift,LevenbergRegularisation,StationaryPoint,
  LocalMinimum,SaddlePoint,FirstOrderOptimality,SecondOrderOptimality,
  StrongConvexity,LocalConvergence,QuadraticConvergence,
  SuperlinearConvergence,BasinOfAttraction,InitialGuess,
  PredictedReduction,ActualReduction,ReductionRatio,
  LinearSystem,LinearSolver,CholeskyFactorisation,LDLFactorisation,
  QRFactorisation,ConjugateGradientMethod,MINRES,GMRES,
  SparseHessian,SparseMatrix,MatrixFreeMethod,HessianVectorProduct,
  AutomaticDifferentiation,NumericalDifferentiation,
  FiniteDifferenceGradient,FiniteDifferenceHessian,
  Scaling,Conditioning,Preconditioner,GradientNorm,StepNorm,
  NewtonDecrement,StoppingCriterion,GradientTolerance,
  StepTolerance,FunctionTolerance,Stagnation,Divergence,
  FloatingPointArithmetic,MachinePrecision,RoundOffError,
  Verification,Benchmarking,ReferenceSolution,Reproducibility
 ]
 EXPAND(NewtonOptimisationFamilies) -> [
  ClassicalNewtonOptimisation,
  DampedAndLineSearchNewton,
  TrustRegionNewton,
  ModifiedAndRegularisedNewton,
  InexactAndTruncatedNewton,
  NewtonCGAndKrylovMethods,
  SparseAndMatrixFreeNewton
 ]
 STOP_CONDITION = "Stop when gradient, Hessian, curvature, linear solve, globalisation, convergence, stopping and validation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Computational Optimisation Ontology"
 RELATED_ONTOLOGIES = [
  "Computational Optimisation Ontology",
  "Optimisation Computation",
  "Nonlinear Optimisation Solver Ontology",
  "Gradient Descent Ontology 2",
  "Newton-Raphson Ontology",
  "Linear Solver Ontology",
  "Conjugate Gradient Ontology",
  "Cholesky Decomposition Ontology",
  "LU Decomposition Ontology",
  "QR Decomposition Ontology",
  "Numerical Linear Algebra Ontology 2",
  "Numerical Differentiation Ontology 2",
  "Numerical Stability Ontology 2",
  "Numerical Convergence Ontology",
  "Floating Point Arithmetic Ontology"
 ]
 ROUTE_RULES = [
  "Newton Optimisation -> Computational Optimisation for parent semantics",
  "Newton Optimisation -> Nonlinear Optimisation Solver for nonlinear objective context",
  "Newton Optimisation -> Gradient Descent for first-order comparison",
  "Newton Optimisation -> Newton-Raphson for stationarity-equation relationship",
  "Newton Optimisation -> Linear Solver for Hessian systems",
  "Newton Optimisation -> Cholesky or LDL factorisation for Hessian solves",
  "Newton Optimisation -> Numerical Differentiation for approximate Hessians",
  "Newton Optimisation -> Numerical Linear Algebra for conditioning and Krylov methods",
  "Newton Optimisation -> Numerical Stability for ill-conditioned Hessian systems",
  "Newton Optimisation -> Numerical Convergence for local quadratic convergence"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible Newton variants, curvature controls or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve gradient-versus-Hessian distinction.",
  "Preserve raw-Newton-versus-globalised-Newton distinction.",
  "Preserve positive-definite-versus-indefinite Hessian distinction.",
  "Preserve exact-versus-inexact linear solve distinction.",
  "Preserve local-model-versus-global-objective distinction.",
  "Preserve stationary-point-versus-local-minimum distinction.",
  "Preserve full-step-versus-damped-or-trust-region distinction.",
  "Preserve machine-readable Newton-optimisation semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Newton optimisation computes steps by solving a Hessian system derived from a local quadratic model of the objective."
 DIRECTION_SUMMARY = "The classical Newton direction solves H p = -g."
 GLOBALISATION_SUMMARY = "Line search, damping and trust regions improve robustness when the local quadratic model is not yet reliable."
 CURVATURE_SUMMARY = "Indefinite or singular Hessians require modification, regularisation or negative-curvature handling."
 CONVERGENCE_SUMMARY = "Near a nondegenerate local minimiser, exact Newton iterations can converge quadratically under standard smoothness assumptions."
 LARGE_SCALE_SUMMARY = "Sparse, inexact and matrix-free Newton methods avoid forming or factorising dense Hessians."
 VALIDATION_SUMMARY = "Newton optimisation should be checked through gradient norms, Hessian curvature, linear-solve accuracy, second-order conditions, convergence rate and reproducibility."
}

FLOW {
 INPUT
 -> detect_Newton_optimisation_context
 -> identify_objective_variables_smoothness_convexity_sparsity_scale_and_accuracy_goal
 -> obtain_gradient_and_Hessian
 -> assess_Hessian_definiteness_conditioning_and_negative_curvature
 -> configure_direct_iterative_sparse_or_matrix_free_Hessian_solver
 -> compute_Newton_or_modified_Newton_direction
 -> apply_full_step_line_search_damping_trust_region_or_regularisation
 -> evaluate_actual_and_predicted_reduction
 -> monitor_gradient_norm_step_norm_Newton_decrement_conditioning_and_progress
 -> adjust_curvature_handling_or_linear_solve_accuracy_if_required
 -> classify_quadratic_superlinear_linear_stagnant_or_divergent_behaviour
 -> validate_second_order_conditions_reference_quality_and_reproducibility
 -> route_to_related_optimisation_nonlinear_solver_gradient_descent_Newton_Raphson_linear_solver_factorisation_differentiation_linear_algebra_stability_and_convergence_ontologies
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Newton Optimisation Ontology defines second-order optimisation using gradient and Hessian information. It supports classical full Newton steps, damped and line-search Newton methods, trust-region Newton, modified and regularised Hessians, modified Cholesky and diagonal shifts, inexact and truncated Newton methods, Newton-CG and Krylov implementations, positive-definite and indefinite Hessian reasoning, negative-curvature handling, sparse Hessians, Hessian-vector products and matrix-free methods, automatic and numerical derivatives, local quadratic and superlinear convergence analysis, scaling, conditioning, stopping criteria, finite precision, verification, benchmarking and reproducibility."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-COMPUTATIONAL-MATHEMATICS-NEWTON-OPTIMISATION-ONTOLOGY-V1"
}

}

END_FLOW