Mathematics Optimization Problem Classes From Config Dim Optimization Classes Ce

Five canonical classes

Optimization problems ask for the best choice among many options, and textbooks sort them into families by what makes them hard: whether the variables are continuous or discrete, whether the constraints are straight lines or curves, whether the data is known or random, and whether decisions happen once or in sequence. A standard five-way classification names linear programming, convex nonlinear programming, integer programming, stochastic programming, and dynamic programming. These five classes cover the main shapes that appear in operations research, from airline scheduling to portfolio choice.

The Recognition Science framework's machine-checked library of formal theorems contains a small structure, OptimizationClassesCert, whose entire content is a proof that the number of these classes is exactly five. The certificate does not define what the classes mean, does not rank them by difficulty, and does not say which problems belong to which class. It records only the count, as a formal object that the library's checker has verified from the definitions. The underlying inductive type names the five classes directly, and a companion theorem, optimizationClass_count, computes the cardinality as five by evaluating the finite set.

In plain terms, the certificate is a formal receipt that says "five," nothing more. It is the kind of small, precise object that a formal library accumulates as a building block: a fact so simple it can be checked by direct computation, yet stated in a form that later, larger theorems can rely on. The library's checker accepts it with no unresolved proofs and no extra axioms, which means the count rests on the definitions alone.

What the certificate does not claim is worth stating plainly. It does not assert that these five classes are the only possible way to organize optimization problems, nor that the classification is exhaustive in any practical sense. It does not claim that every optimization problem falls into exactly one of the five, and it does not assign any problem to a class. The certificate is a statement about a formal definition, not about the world of algorithms or applications.

structure OptimizationClassesCert where
  five_classes : Fintype.card OptimizationClass = 5

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

What this page does not claim

The certificate does not define the content or meaning of the five optimization classes. The certificate does not claim the five classes are exhaustive or that every optimization problem falls into exactly one. The certificate does not assign any specific optimization problem to a class.

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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