Mathematics Optimization Theory From Rs Local Minimum

The local minimum

In classical optimization, a local minimum is a candidate solution that is better than every nearby alternative, but not necessarily the best solution in the whole search space. The global minimum is the single best point. The local minimum is the second-best kind of answer: it is a point from which no small step improves the outcome, yet a larger move might find something better. This distinction is the backbone of numerical methods, where algorithms often settle into a local valley without ever reaching the global floor.

Optimization theory classically recognizes five canonical problem types: linear, nonlinear, combinatorial, convex, and stochastic. Each has its own standard methods and pitfalls. The ledger, a discrete record of recognition events, is the framework's name for the cost function J, and the framework treats all optimization as the minimization of this single cost function.

In Recognition Science, the framework's library of machine-checked formal theorems proves that the cost function J has exactly one global minimum: J = 0, reached at the point r = 1. The declaration local_minimum proves the companion fact: for any positive r not equal to 1, the cost is strictly positive, meaning J(r) > 0. In plain language, the only point where the cost vanishes is the equilibrium point r = 1; every other positive point has a positive cost, so it is at best a local minimum, never a global one.

This is a theorem about the cost function's shape, not a recipe for finding local minima. The declaration does not claim that every local minimum is reachable by any particular algorithm, nor that the cost function has no other critical points. It only states the strict positivity of the cost away from the unique global minimum. The five problem types are counted by a separate theorem, and the local minimum result does not depend on that count.

The practical consequence is a clean separation: the global minimum is unique and known, and every other point carries a positive cost. An optimizer that lands anywhere other than r = 1 knows it has not finished. The theorem gives a certificate, a formal guarantee, that the search is incomplete until the cost reaches zero.

/-- Local minimum: J > 0. -/
theorem local_minimum {r : ℝ} (hr : 0 < r) (hne : r ≠ 1) :
    0 < Jcost r := Jcost_pos_of_ne_one r hr hne

/-- Global minimum: J = 0. -/
theorem global_minimum : Jcost 1 = 0 := Jcost_unit0

theorem optimizationProblemTypeCount : Fintype.card OptimizationProblemType = 5 := by decide

What this page does not claim

The declaration does not prove that any particular algorithm converges to a local or global minimum. The declaration does not claim that the cost function has no other stationary points besides r = 1. The declaration does not address the existence of local minima in the five canonical problem types.

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

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