Computational Optimisation Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Computational Optimisation Ontology within the Kireth Research Semantic Library. Computational optimisation concerns algorithmic methods for solving optimisation problems numerically, including unconstrained and constrained problems, convex and nonconvex objectives, linear and nonlinear programming, gradient-based and derivative-free methods, line searches, trust regions, penalties, barriers, duality, KKT systems, stochastic methods, large-scale computation, convergence, conditioning, stopping criteria, verification and reproducibility.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = ComputationalOptimisation
 ENTITY_CLASS = AlgorithmicOptimisationFramework
 PRIMARY_CONTEXT = OptimisationComputation
 DEFINITION = "Computational optimisation is the numerical solution of optimisation problems through iterative or discrete algorithms that seek decision variables minimising or maximising an objective subject to declared constraints."
 SYMBOLIC_FORMS = [
  "x_star=argmin_x f(x)",
  "x_star=argmax_x f(x)",
  "min_x f(x) subject_to g(x)<=0, h(x)=0",
  "r_KKT(x,lambda,mu)=0"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for optimisation problem classification, numerical algorithms, derivative information, feasibility, optimality, globalisation, convergence, conditioning, stopping and verification."
}

SCOPE {
 INCLUDE = [
  ComputationalOptimisation,OptimisationAlgorithm,OptimisationProblem,
  ObjectiveFunction,DecisionVariable,SearchSpace,FeasibleSet,
  Constraint,EqualityConstraint,InequalityConstraint,BoundConstraint,
  LinearConstraint,NonlinearConstraint,InitialGuess,InitialPoint,
  UnconstrainedOptimisation,ConstrainedOptimisation,
  LinearProgramming,QuadraticProgramming,NonlinearProgramming,
  ConvexOptimisation,NonconvexOptimisation,SmoothOptimisation,
  NonsmoothOptimisation,DiscreteOptimisation,IntegerProgramming,
  MixedIntegerProgramming,CombinatorialOptimisation,
  GlobalOptimisation,LocalOptimisation,MultiobjectiveOptimisation,
  Gradient,Jacobian,Hessian,Subgradient,DirectionalDerivative,
  AutomaticDifferentiation,NumericalDifferentiation,
  FiniteDifferenceGradient,FiniteDifferenceHessian,
  FirstOrderMethod,SecondOrderMethod,ZeroOrderMethod,
  GradientDescent,SteepestDescent,NewtonOptimisation,
  QuasiNewtonMethod,BFGS,DFP,LBFGS,
  GaussNewtonMethod,LevenbergMarquardtMethod,
  ConjugateGradientOptimisation,CoordinateDescent,
  BlockCoordinateDescent,LineSearch,ExactLineSearch,
  InexactLineSearch,BacktrackingLineSearch,ArmijoCondition,
  WolfeCondition,StrongWolfeCondition,TrustRegion,
  TrustRegionRadius,DoglegMethod,TruncatedNewtonMethod,
  DerivativeFreeOptimisation,NelderMeadMethod,PowellMethod,
  PatternSearch,DirectSearch,PenaltyMethod,BarrierMethod,
  LogBarrier,AugmentedLagrangian,LagrangeMultiplier,
  Lagrangian,DualFunction,DualProblem,PrimalProblem,
  WeakDuality,StrongDuality,ComplementarySlackness,
  KarushKuhnTuckerConditions,KKTConditions,
  InteriorPointMethod,ActiveSetMethod,SequentialQuadraticProgramming,
  SQP,ProjectedGradient,ProximalMethod,ProximalGradient,
  AlternatingDirectionMethodOfMultipliers,ADMM,
  SimplexMethod,RevisedSimplexMethod,BranchAndBound,
  BranchAndCut,CuttingPlaneMethod,DynamicProgramming,
  StochasticOptimisation,StochasticGradientDescent,SGD,
  MiniBatchGradient,Momentum,NesterovAcceleration,
  AdaGrad,RMSProp,Adam,VarianceReduction,
  BayesianOptimisation,SurrogateOptimisation,
  SimulatedAnnealing,GeneticAlgorithm,ParticleSwarmOptimisation,
  LargeScaleOptimisation,SparseOptimisation,DistributedOptimisation,
  ParallelOptimisation,MatrixFreeOptimisation,
  DecompositionMethod,Preconditioning,Scaling,VariableScaling,
  ObjectiveScaling,ConstraintScaling,Conditioning,Sensitivity,
  ConstraintViolation,PrimalResidual,DualResidual,
  StationarityResidual,KKTResidual,OptimalityResidual,
  LocalMinimum,GlobalMinimum,LocalMaximum,GlobalMaximum,
  StationaryPoint,CriticalPoint,SaddlePoint,
  FirstOrderOptimality,SecondOrderOptimality,
  ConvexSet,ConvexFunction,StrongConvexity,LipschitzGradient,
  Regularisation,L1Regularisation,L2Regularisation,
  Sparsity,PenaltyParameter,BarrierParameter,StepSize,
  LearningRate,Iteration,MaximumIterations,
  FunctionEvaluationCount,GradientEvaluationCount,
  HessianEvaluationCount,StoppingCriterion,
  GradientTolerance,StepTolerance,FunctionTolerance,
  ConstraintTolerance,KKT_Tolerance,Convergence,
  ConvergenceRate,SublinearConvergence,LinearConvergence,
  SuperlinearConvergence,QuadraticConvergence,
  Stagnation,Divergence,Failure,Success,Warning,
  FloatingPointArithmetic,MachinePrecision,RoundOffError,
  Overflow,Underflow,Verification,Benchmarking,
  ReferenceSolution,Reproducibility
 ]
 EXCLUDE_AS_PRIMARY = [
  SymbolicOptimisationWithoutNumericalAlgorithm,
  PureEquationSolvingWithoutObjectiveFunction,
  StatisticalInferenceWithoutOptimisationSemantics,
  SearchHeuristicWithoutObjectiveOrDecisionVariableSemantics
 ]
 DISAMBIGUATION_RULE = "When an optimisation problem is solved through explicit computational algorithms, numerical updates, discrete search or iterative optimality calculations, interpret the context as computational optimisation."
}

CORE_CONCEPTS {
 ComputationalOptimisation = "Algorithmic numerical solution of optimisation problems."
 SearchDirection = "Direction used to generate a candidate optimisation step."
 LineSearch = "Procedure choosing a step length along a search direction."
 TrustRegion = "Framework constraining a local model step to a region of confidence."
 Feasibility = "Satisfaction of all declared optimisation constraints."
 OptimalityResidual = "Measure of departure from first- or second-order optimality conditions."
 KKTConditions = "First-order necessary optimality conditions for constrained problems under suitable regularity."
 Globalisation = "Mechanism that improves robustness of locally derived steps."
}

RELATIONS {
 ComputationalOptimisation --is_child_of--> OptimisationComputation
 ComputationalOptimisation --solves--> OptimisationProblem
 OptimisationProblem --has--> ObjectiveFunction
 OptimisationProblem --has_optional--> Constraint
 Constraint --defines--> FeasibleSet
 GradientDescent --uses--> Gradient
 NewtonOptimisation --uses--> Hessian
 QuasiNewtonMethod --approximates--> HessianOrInverseHessian
 BFGS --is_type_of--> QuasiNewtonMethod
 LBFGS --is_type_of--> QuasiNewtonMethod
 LineSearch --controls--> StepSize
 TrustRegion --controls--> CandidateStep
 Lagrangian --combines--> ObjectiveAndConstraints
 KKTConditions --characterise_under_assumptions--> ConstrainedStationarity
 InteriorPointMethod --uses--> BarrierMethod
 SQP --uses--> QuadraticProgrammingSubproblem
 ProximalGradient --uses--> ProximalOperator
 ADMM --uses--> AugmentedLagrangian
 StochasticGradientDescent --uses--> StochasticGradient
 Verification --checks--> OptimisationResult
}

OPERATORS {
 MINIMISE(f,X) := argmin_x_in_X f(x)
 MAXIMISE(f,X) := argmax_x_in_X f(x)
 GRADIENT_STEP(x,g,alpha) := x_next=x-alpha*g
 NEWTON_STEP(x,g,H) := solve(H*p=-g); x_next=x+p
 PROJECTED_STEP(x,g,alpha,C) := x_next=Projection_C(x-alpha*g)
 LINE_SEARCH(phi,p) := choose_alpha_satisfying_declared_conditions
 TRUST_REGION_SUBPROBLEM(g,H,Delta) := min_p g^T*p+0.5*p^T*H*p subject_to ||p||<=Delta
 LAGRANGIAN(x,lambda,mu) := f(x)+sum_i lambda_i*g_i(x)+sum_j mu_j*h_j(x)
 KKT_RESIDUAL(x,lambda,mu) := combine(stationarity,primal_feasibility,dual_feasibility,complementarity)
}

AXIOMS {
 A1 = "Computational optimisation requires an objective and a declared search or feasible domain."
 A2 = "Feasibility and optimality are distinct for constrained problems."
 A3 = "Local optimality and global optimality are distinct."
 A4 = "Derivative quality influences the reliability of derivative-based optimisation methods."
 A5 = "Line-search and trust-region methods globalise local models through different mechanisms."
 A6 = "Convexity can convert local optimality conditions into global guarantees."
 A7 = "Scaling and conditioning materially affect numerical optimisation."
 A8 = "Optimisation results require explicit stopping, optimality, feasibility and reproducibility validation."
}

REASONING {
 R1_CONTEXT = identify(objective,variables,domain,constraints,smoothness,convexity,discreteness,noise,accuracy_goal)
 R2_CLASSIFY = classify(unconstrained,constrained,linear,quadratic,nonlinear,convex,nonconvex,discrete,stochastic_or_multiobjective)
 R3_DERIVATIVES = determine(analytic,automatic_differentiation,numerical_or_derivative_free_information)
 R4_METHOD = select(first_order,second_order,line_search,trust_region,interior_point,SQP,proximal,derivative_free,stochastic_or_global_method)
 R5_CONFIGURE = set(initial_guess,scaling,tolerances,step_controls,penalty_barrier_or_regularisation_parameters)
 R6_STEP = compute(search_direction_or_candidate_update)
 R7_GLOBALISE = apply(line_search,trust_region,projection,penalty,barrier_or_filter_if_needed)
 R8_MONITOR = track(objective,gradient_norm,constraint_violation,KKT_residual,step_size_and_progress)
 R9_TERMINATE = classify(convergence,stagnation,divergence_or_resource_limit)
 R10_VALIDATE = check(feasibility,optimality_conditions,sensitivity,reference_quality_and_reproducibility)
}

CONSTRAINTS {
 C1 = "A stationary point need not be a local minimum in a nonconvex problem."
 C2 = "Local optimisation does not guarantee a global optimum."
 C3 = "Poor scaling can degrade search directions and stopping tests."
 C4 = "Finite-difference derivatives balance truncation and round-off error."
 C5 = "Indefinite Hessians can produce non-descent Newton directions."
 C6 = "Penalty and barrier parameters can create severe ill-conditioning."
 C7 = "Stochastic gradients introduce noise into convergence assessment."
 C8 = "Derivative-free methods can require many objective evaluations in high dimensions."
 C9 = "Resource-limit termination is not equivalent to convergence."
 C10 = "Small objective change alone does not establish feasibility or first-order optimality."
}

SEMANTIC_EQUIVALENCE {
 "ComputationalOptimisation" <=> "NumericalOptimisation"
 "GradientDescent" <=> "SteepestDescentInEuclideanMetric"
 "LBFGS" <=> "LimitedMemoryBFGS"
 "KKTConditions" <=> "KarushKuhnTuckerConditions"
 "SGD" <=> "StochasticGradientDescent"
 "ADMM" <=> "AlternatingDirectionMethodOfMultipliers"
}

VALIDATION {
 V1 = "Objective function and optimisation direction are explicit."
 V2 = "Decision variables, bounds and constraints are documented."
 V3 = "Initialisation is documented."
 V4 = "Derivative source and optimisation algorithm are documented."
 V5 = "Scaling and preconditioning are documented where relevant."
 V6 = "Stopping criteria and tolerances are explicit."
 V7 = "Constraint feasibility is checked for constrained problems."
 V8 = "Gradient, KKT or equivalent optimality residuals are reported where applicable."
 V9 = "Reference or benchmark solutions are used where feasible."
 V10 = "Results are reproducible under documented arithmetic and algorithm settings."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 COMPUTATIONAL_OPTIMISATION_DEFINITION = 0.99
 FIRST_ORDER_REASONING = 0.99
 SECOND_ORDER_REASONING = 0.99
 CONSTRAINED_REASONING = 0.99
 KKT_REASONING = 0.99
 DERIVATIVE_FREE_REASONING = 0.98
 STOCHASTIC_REASONING = 0.98
 GLOBAL_REASONING = 0.97
 POLICY = "Reduce confidence when objective, constraints, convexity, derivative quality, scaling, initialisation, stopping criteria or stochastic noise are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,OntologyGrounding,ComputationalOptimisationReasoning,
  FirstOrderOptimisationReasoning,SecondOrderOptimisationReasoning,
  GradientDescentReasoning,NewtonOptimisationReasoning,
  QuasiNewtonReasoning,LineSearchReasoning,TrustRegionReasoning,
  ConstrainedOptimisationReasoning,KKTReasoning,
  InteriorPointReasoning,SQPReasoning,ProximalReasoning,
  ADMMReasoning,DerivativeFreeReasoning,
  StochasticOptimisationReasoning,GlobalOptimisationReasoning,
  LargeScaleOptimisationReasoning,StoppingCriterionReasoning,
  VerificationReasoning,KnowledgeGraphConstruction,
  RetrievalAugmentation,RecursiveInference
 ]
 NORMALISE {
  "computational optimisation" -> ComputationalOptimisation
  "computational optimization" -> ComputationalOptimisation
  "numerical optimisation" -> ComputationalOptimisation
  "L-BFGS" -> LBFGS
  "KKT" -> KKTConditions
  "SGD" -> StochasticGradientDescent
  "ADMM" -> AlternatingDirectionMethodOfMultipliers
 }
}

EXECUTION_RULES {
 E1 = "If an optimisation problem is solved numerically through an explicit algorithm, activate ComputationalOptimisationOntology."
 E2 = "Classify constraints, smoothness, convexity, discreteness and stochasticity before selecting the algorithm."
 E3 = "Use derivative information only when its source and quality are sufficiently reliable."
 E4 = "Globalise local model steps with line search, trust region, projection or related controls as required."
 E5 = "For constrained smooth problems, monitor feasibility and KKT residuals."
 E6 = "For large-scale problems, exploit sparsity, matrix-free operations and preconditioning."
 E7 = "For noisy or stochastic objectives, use noise-aware progress and stopping criteria."
 E8 = "Distinguish successful convergence from stagnation, failure and resource-limit termination."
 E9 = "Validate objective quality, feasibility, optimality residuals, sensitivity and reproducibility."
}

RECURSION {
 ENTRY = ComputationalOptimisation
 EXPAND(ComputationalOptimisation) -> [
  OptimisationAlgorithm,OptimisationProblem,ObjectiveFunction,
  DecisionVariable,SearchSpace,FeasibleSet,Constraint,
  EqualityConstraint,InequalityConstraint,BoundConstraint,
  UnconstrainedOptimisation,ConstrainedOptimisation,
  LinearProgramming,QuadraticProgramming,NonlinearProgramming,
  ConvexOptimisation,NonconvexOptimisation,SmoothOptimisation,
  NonsmoothOptimisation,IntegerProgramming,MixedIntegerProgramming,
  GlobalOptimisation,StochasticOptimisation,Gradient,Jacobian,
  Hessian,Subgradient,AutomaticDifferentiation,
  NumericalDifferentiation,FirstOrderMethod,SecondOrderMethod,
  GradientDescent,NewtonOptimisation,QuasiNewtonMethod,BFGS,LBFGS,
  GaussNewtonMethod,LevenbergMarquardtMethod,
  CoordinateDescent,LineSearch,ArmijoCondition,WolfeCondition,
  TrustRegion,DoglegMethod,TruncatedNewtonMethod,
  DerivativeFreeOptimisation,NelderMeadMethod,PowellMethod,
  PatternSearch,PenaltyMethod,BarrierMethod,AugmentedLagrangian,
  Lagrangian,DualProblem,KKTConditions,InteriorPointMethod,
  ActiveSetMethod,SQP,ProjectedGradient,ProximalGradient,ADMM,
  SimplexMethod,BranchAndBound,CuttingPlaneMethod,
  StochasticGradientDescent,Momentum,NesterovAcceleration,
  AdaGrad,RMSProp,Adam,BayesianOptimisation,
  SimulatedAnnealing,GeneticAlgorithm,ParticleSwarmOptimisation,
  LargeScaleOptimisation,SparseOptimisation,DistributedOptimisation,
  MatrixFreeOptimisation,Scaling,Conditioning,ConstraintViolation,
  PrimalResidual,DualResidual,StationarityResidual,KKTResidual,
  OptimalityCondition,LocalMinimum,GlobalMinimum,SaddlePoint,
  Regularisation,StoppingCriterion,Convergence,Stagnation,
  FloatingPointArithmetic,Verification,Benchmarking,Reproducibility
 ]
 EXPAND(ComputationalOptimisationFamilies) -> [
  GradientBasedOptimisation,
  NewtonAndQuasiNewtonOptimisation,
  ConstrainedOptimisationAlgorithms,
  ConvexOptimisationAlgorithms,
  DerivativeFreeOptimisation,
  StochasticOptimisation,
  DiscreteAndGlobalOptimisation,
  LargeScaleAndDistributedOptimisation
 ]
 STOP_CONDITION = "Stop when objective, variables, constraints, derivative information, algorithm, globalisation, convergence, optimality and verification are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Optimisation Computation"
 RELATED_ONTOLOGIES = [
  "Optimisation Computation",
  "Numerical Methods",
  "Numerical Analysis Ontology 2",
  "Numerical Differentiation Ontology 2",
  "Equation Solver Ontology",
  "Linear Solver Ontology",
  "Nonlinear Solver Ontology",
  "Numerical Linear Algebra Ontology 2",
  "Numerical Error Ontology",
  "Numerical Stability Ontology 2",
  "Numerical Convergence Ontology",
  "Floating Point Arithmetic Ontology",
  "Scientific Computing Ontology 2",
  "Artificial Intelligence Mathematics"
 ]
 ROUTE_RULES = [
  "Computational Optimisation -> Optimisation Computation for parent semantics",
  "Computational Optimisation -> Numerical Differentiation for finite-difference gradients and Hessians",
  "Computational Optimisation -> Equation Solver for stationarity and KKT systems",
  "Computational Optimisation -> Linear Solver for Newton, SQP and interior-point subproblems",
  "Computational Optimisation -> Nonlinear Solver for nonlinear optimality systems",
  "Computational Optimisation -> Numerical Linear Algebra for Hessians, factorisations and preconditioning",
  "Computational Optimisation -> Numerical Analysis for convergence and conditioning",
  "Computational Optimisation -> Scientific Computing for scalable implementation",
  "Computational Optimisation -> Artificial Intelligence Mathematics for learning and training optimisation"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible optimisation algorithms or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve objective-versus-constraint distinction.",
  "Preserve feasible-versus-optimal distinction.",
  "Preserve local-versus-global optimum distinction.",
  "Preserve first-order-versus-second-order algorithm distinction.",
  "Preserve derivative-based-versus-derivative-free distinction.",
  "Preserve deterministic-versus-stochastic distinction.",
  "Preserve continuous-versus-discrete optimisation distinction.",
  "Preserve machine-readable computational-optimisation semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Computational optimisation solves optimisation problems numerically using iterative, derivative-based, derivative-free, stochastic or discrete algorithms."
 CLASSIFICATION_SUMMARY = "Algorithm choice depends on constraints, smoothness, convexity, dimensionality, discreteness and stochasticity."
 DERIVATIVE_SUMMARY = "Gradient, Jacobian, Hessian and subgradient information support increasingly rich local models."
 GLOBALISATION_SUMMARY = "Line search and trust-region strategies improve robustness of local optimisation steps."
 CONSTRAINED_SUMMARY = "KKT residuals, duality, interior-point, active-set and SQP methods govern constrained optimisation semantics."
 LARGE_SCALE_SUMMARY = "Sparse, matrix-free, stochastic and distributed algorithms support large computational optimisation problems."
 VALIDATION_SUMMARY = "Optimisation results should be checked for objective quality, feasibility, optimality residuals, convergence, sensitivity and reproducibility."
}

FLOW {
 INPUT
 -> detect_computational_optimisation_context
 -> identify_objective_variables_domain_constraints_smoothness_convexity_discreteness_noise_and_accuracy_goal
 -> classify_optimisation_problem
 -> determine_derivative_information
 -> select_optimisation_algorithm
 -> configure_initialisation_scaling_tolerances_and_globalisation
 -> compute_candidate_step_or_search_update
 -> apply_line_search_trust_region_projection_penalty_barrier_or_related_control
 -> evaluate_objective_constraints_optimality_residuals_and_progress
 -> monitor_convergence_stagnation_divergence_and_resource_limits
 -> assess_conditioning_sensitivity_roundoff_and_stochastic_noise
 -> validate_feasibility_optimality_reference_quality_and_reproducibility
 -> route_to_related_optimisation_numerical_methods_differentiation_equation_solver_linear_algebra_analysis_stability_convergence_scientific_computing_and_AI_mathematics_ontologies
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Computational Optimisation Ontology defines algorithmic numerical optimisation across unconstrained, constrained, linear, quadratic, nonlinear, convex, nonconvex, discrete and stochastic problems. It supports gradient descent, Newton and quasi-Newton methods, BFGS and L-BFGS, Gauss-Newton and Levenberg-Marquardt methods, coordinate descent, line searches, trust regions, derivative-free methods, penalties, barriers, augmented Lagrangians, KKT systems, interior-point, active-set and SQP methods, projected and proximal methods, ADMM, simplex, branch-and-bound and cutting-plane methods, stochastic gradients, momentum, AdaGrad, RMSProp and Adam, global and surrogate optimisation, sparse, matrix-free and distributed optimisation, scaling, conditioning, stopping criteria, finite precision, verification, benchmarking and reproducibility."
}

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

}

END_FLOW