Iterated local search acceptance rule

An iterated local search acceptance rule chooses the next current solution after a perturbation has been followed by local search. Let the current locally improved solution be $s$ and the new locally improved candidate be $q$, with a minimized objective function $f$. The decision concerns these search states, not just the potentially poor intermediate output of the local-search perturbation.^[ref]

One rule accepts only improvement:The paper calls this policy Better. Its current costs cannot increase, which follows immediately from the two cases. That property does not establish that the search will reach a global minimum. The RW rule instead retains the latest candidate regardless of cost, allowing movement among locally improved solutions.^[ref]

For an original numerical comparison, suppose $f(s)=10$. A perturbation temporarily raises the cost to18, and subsequent local search lowers it to12. Better rejects the candidate with cost12, while RW accepts it; neither decision is made by comparing10 with the intermediate18. If the same procedure instead reaches cost8, both accept.

A simulated annealing-type option accepts a worse candidate with probability $\exp [-(f(q)-f(s))/T]$, with $T>0$. At cost difference2 and $T=2$, this probability is $e^{-1}\simeq 0.368$. The temperature is a parameter of the rule, not a guarantee that a particular finite run escapes every trap.^[ref]

A permissive current-state rule can retain a candidate worse than an earlier result. Keeping a separate best-so-far solution preserves that result while allowing exploration. Report both the current-state acceptance policy and the best-result bookkeeping when comparing search runs.