The 2-epistasis of fitness functions

Abstract

In this note about Genetic Algorithms(GA-), we study the2-epistasisof a fitness function over a search space. This concept is a natural generalisation of that ofepistasis, previously considered by Davidor in 1991 Suys and Verschoren in 1996 and Van Hove and Verschoren in 1994 for example. We completely characterise fitness functions whose 2-epistasis is minimal: these are exactly the second order functions. The validity of 2-epistasis as a measure of hardness with respect to genetic algorithms is checked over some classical laboratory functions. Finally, we obtain an upper bound of the maximal value of the 2-epistasis when we restrict attention to non-negative functions.

Higher Epistasis in Genetic Algorithms

“…Theorem 3.11 extends the analogous result for classical normalized epistasis and 2-epistasis established in [8] and [6], respectively.…”

“…This section is devoted to constructing, for any pair of positive integers , k, the matrices G ,k which are a natural generalization of those obtained in [13] for normalized epistasis, and of those obtained in [6,10] for 2-epistasis. These matrices allow normalized higher epistasis to be introduced in an algebraic form consistent with that in [6,13].…”

“…to be the genic excess of the string s ∈ . Then, it follows that E(s) + f = (Ef) s may be considered as the predicted genic value of s and, finally, just as in the 'classical' case k = 1 and the case k = 2 (see [1] and [6], respectively), the difference e s = f (s) − (Ef) s measures the k-epistasis of f for arbitrary k. [12]…”

“…The relation between the algebraic definition of higher epistasis and Davidor's 'classical' approach [1] is the subject of Section 4. We show, in particular, how our set-up includes the 2-epistasis studied in [6] as a special case.…”