Reference · The Applied Higher-Order Functions catalogue

rainbow.algorithms.proofs.ahof.column-generation

The Applied Higher-Order Functions cataloguesrc/rainbow/algorithms/proofs/ahof/column_generation.clj1 definitions, 3 theorems

AHoF O-family Column Generation: solve a large linear program with too many variables to enumerate by working with a small RESTRICTED master problem and generating new columns only as needed. The pricing subproblem asks whether any not-yet-included column would improve the objective. Classically a column can improve a minimization objective when its reduced cost (cost - dual, with dual the price of the constraints it serves) is strictly negative, i.e. cost < dual. The test here is non-strict: worth-adding? cost dual = decide (cost <= dual) over exact rationals, so it accepts every column of NONPOSITIVE reduced cost, including cost = dual (reduced cost zero, e.g. cost 5 with dual 5), which does not strictly improve the objective. Defined: the pricing test. Proved native and axiom-free, as worked instances: a column of cost 3 with dual value 5 prices favorably, a column of cost 7 with dual value 5 does not, and the worked certificate 3 <= 5 (native nnrat/at-most). The classical guarantees (an added column improves or maintains the objective; when no column has strictly negative reduced cost the current basis is optimal) are not proved here. The subproblem solver and the master LP are executable. Requires nnrat/at-most, audit/decide-nnrat-at-most, nnrat/decide-at-most-true-sound (all in the D63 checkpoint). Scope (measured by the kernel): this module has 3 worked instances (statements about specific values). None of its theorems is a guarantee about the algorithm for all inputs; `lein verify rainbow.algorithms.proofs.ahof.column-generation` prints this classification.

column-generation/worth-adding? :
  (x₁ x₂ : ℚ≥0)
  → data/Bool

column-generation/prices-favorably :
  → ce-fal/elite? 3 5 = data/true

column-generation/added-column-improves :
  → 3 ≤ 5

column-generation/pricing-converged :
  → ce-fal/elite? 7 5 = data/false