Reference · The Applied Higher-Order Functions catalogue

rainbow.algorithms.proofs.ahof.chebyshev-scalarization

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

AHoF multi-objective method: weighted-Chebyshev (max-norm) scalarization. A multi-objective problem is reduced to a single scalar by taking the maximum of the (weighted) objective deviations: scalarize [v0, v1, ...] = list-maximum [v0, v1, ...]. Minimizing this max-norm drives every objective toward its target and, unlike a weighted SUM, can reach every point of a non-convex Pareto front. The defining property is domination: the scalarized value is an upper bound on every component, so no single objective exceeds it. Proved native and axiom-free: the scalarization is exactly the proved list-maximum fold; a worked scalarization; and that the scalarized value dominates each component (component <= scalarized as native nnrat/at-most). The weighting and the target vector are executable concerns; what is proved is the max-norm scalarization and its domination. Requires list-maximum/maximum, nnrat/at-most, audit/decide-nnrat-at-most, nnrat/decide-at-most-true-sound, and 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.chebyshev-scalarization` prints this classification.

Depends onalgorithms.proofs.list-maximum

chebyshev/scalarize :
  (x₁ : List ℚ≥0)
  → ℚ≥0

chebyshev/scalarize-example :
  → chebyshev/scalarize [3, 7, 2] = 7

chebyshev/dominates-first :
  → 3 ≤ 7

chebyshev/dominates-second :
  → 7 ≤ 7

chebyshev/dominates-third :
  → 2 ≤ 7