Mathematics Optimization Problem Classes From Config Dim Optimization Class

Five canonical classes

Optimization is the branch of mathematics that chooses the best option from a set of alternatives, subject to constraints. The field organizes its problems into families by shape and method. The declaration OptimizationClass fixes one such taxonomy at exactly five members: linear, convex nonlinear, integer, stochastic, and dynamic. The first two describe the objective function's form, the third restricts variables to whole numbers, the fourth brings in randomness, and the fifth sequences decisions over time. This five-way split matches the standard operations-research survey of the discipline.

The declaration is an inductive type, a way of saying the list is closed: these five names are the only members, with no hidden sixth case. A companion theorem, optimizationClass_count, proves the count is five by deciding the matter directly. That proof is a computation, not an argument from assumptions. A small certificate structure packages the count so other parts of the library can rely on it without rechecking. The whole file carries no unresolved proofs and no added axioms, meaning a machine checked every step.

In Recognition Science, the number five is not arbitrary. The framework's forcing chain derives the dimension of configuration space as D = 5 from the recognition cost function, and this declaration connects that derived dimension to the classical taxonomy. The framework models the five optimization classes as the natural partition of problem types at that dimension. The connection is a definitional choice, not a proof that real-world optimization problems must fall into these five buckets.

What the declaration does not do is more important than what it names. It does not prove that every optimization problem belongs to one of these five classes, nor that the classes are disjoint. It does not rank the classes by difficulty or importance. It does not claim that the five names exhaust all possible optimization methods, only that this taxonomy has five members. The theorem certifies the count of a chosen list; it does not justify the choice of list.

inductive OptimizationClass where
  | linear
  | convexNonlinear
  | integer
  | stochastic
  | dynamic
  deriving DecidableEq, Repr, BEq, Fintype

theorem optimizationClass_count : Fintype.card OptimizationClass = 5 := by decide

structure OptimizationClassesCert where
  five_classes : Fintype.card OptimizationClass = 5

inductive OptimizationClass where
  | linear
  | convexNonlinear
  | integer
  | stochastic
  | dynamic
  deriving DecidableEq, Repr, BEq, Fintype

What this page does not claim

The declaration does not prove that every optimization problem belongs to one of these five classes. The declaration does not establish that the five classes are disjoint or exhaustive of all possible optimization methods. The declaration does not rank the classes by difficulty or practical importance.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Mathematics/OptimizationProblemClassesFromConfigDim.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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