Number theory
While the word "algorithm" goes back only to certain readers of al-Khwārizmī, careful descriptions of methods of solution are older than proofs: such methods (that is, algorithms) are as old as any recognisable mathematics—ancient Egyptian, Babylonian, Vedic, Chinese—whereas proofs appeared only with the Greeks of the classical period.
An interesting early case is that of what we now call the Euclidean algorithm. In its basic form (namely, as an algorithm for computing the greatest common divisor) it appears as Proposition 2 of Book VII in "Elements", together with a proof of correctness. However, in the form that is often used in number theory (namely, as an algorithm for finding integer solutions to an equation formula_78,
or, what is the same, for finding the quantities whose existence is assured by the Chinese remainder theorem) it first appears in the works of Āryabhaṭa (5th–6th century CE) as an algorithm called
"kuṭṭaka" ("pulveriser"), without a proof of correctness.
Hyper-heuristic
The term "hyperheuristics" was first coined in a 2000 publication by Cowling and Soubeiga, who used it to describe the idea of "heuristics to choose heuristics". They used a "choice function" machine learning approach which trades off exploitation and exploration in choosing the next heuristic to use. Subsequently, Cowling, Soubeiga, Kendall, Han, Ross and other authors investigated and extended this idea in areas such as evolutionary algorithms, and pathological low level heuristics. The first journal article to use the term appeared in 2003. The origin of the idea (although not the term) can be traced back to the early 1960s and was independently re-discovered and extended several times during the 1990s. In the domain of Job Shop Scheduling, the pioneering work by Fisher and Thompson, hypothesized and experimentally proved, using probabilistic learning, that combining scheduling rules (also known as priority or dispatching rules) was superior than any of the rules taken separately. Although the term was not then in use, this was the first "hyper-heuristic" paper. Another root inspiring the concept of hyper-heuristics comes from the field of artificial intelligence. More specifically, it comes from work on automated planning systems, and its eventual focus towards the problem of learning control knowledge. The so-called COMPOSER system, developed by Gratch et al., was used for controlling satellite communication schedules involving a number of earth-orbiting satellites and three ground stations. The system can be characterized as a hill-climbing search in the space of possible control strategies.