  Gray-box optimization leverages the information available about the
mathematical structure of an optimization problem to design efficient search
operators. Efficient hill climbers and crossover operators have been proposed
in the domain of pseudo-Boolean optimization and also in some permutation
problems. However, there is no general rule on how to design these efficient
operators in different representation domains. This paper proposes a general
framework that encompasses all known gray-box operators for combinatorial
optimization problems. The framework is general enough to shed light on the
design of new efficient operators for new problems and representation domains.
We also unify the proofs of efficiency for gray-box hill climbers and
crossovers and show that the mathematical property explaining the speed-up of
gray-box crossover operators, also explains the efficient identification of
improving moves in gray-box hill climbers. We illustrate the power of the new
framework by proposing an efficient hill climber and crossover for two related
permutation problems: the Linear Ordering Problem and the Single Machine Total
Weighted Tardiness Problem.
