Reference · The Applied Higher-Order Functions catalogue

rainbow.algorithms.proofs.ahof.pareto

The Applied Higher-Order Functions cataloguesrc/rainbow/algorithms/proofs/ahof/pareto.clj1 definitions, 4 theorems

AHoF multiobjective optimization -- Pareto dominance, the partial order that defines the Pareto front. For a minimization problem with two objectives, a solution a weakly dominates b when a is no worse than b in every objective: dominates? a1 a2 b1 b2 = (a1 <= b1) and (a2 <= b2) expressed as nested if-data over the decidable rational order (no boolean connective needed). Proved native and axiom-free, as worked instances: (2,3) dominates (5,4); the worked certificates 2 <= 5 and 3 <= 4 (native nnrat/at-most); and (6,3) does not dominate (5,4). The soundness of the test for all inputs, and of a Pareto-front filter built on it, is not proved here. Extending to k objectives and extracting the non-dominated front are executable concerns. Requires nnrat/at-most, audit/decide-nnrat-at-most, nnrat/decide-at-most-true-sound, and logic/if-data (all in the D63 checkpoint). Scope (measured by the kernel): this module has 4 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.pareto` prints this classification.

pareto/dominates? :
  (x₁ x₂ x₃ x₄ : ℚ≥0)
  → data/Bool

pareto/dominates-example :
  → pareto/dominates? 2 3 5 4 = data/true

pareto/dominated-in-objective-1 :
  → 2 ≤ 5

pareto/dominated-in-objective-2 :
  → 3 ≤ 4

pareto/non-dominance-example :
  → pareto/dominates? 6 3 5 4 = data/false