Tunably Rugged Landscapes With Known Maximum and Minimum

Abstract

We propose NM landscapes as a new class of tunably rugged benchmark problems. NM landscapes are well defined on alphabets of any arity, including both discrete and real-valued alphabets, include epistasis in a natural and transparent manner, are proven to have known value and location of the global maximum and, with some additional constraints, are proven to also have a known global minimum. Empirical studies are used to illustrate that, when coefficients are selected from a recommended distribution, the ruggedness of NM landscapes is smoothly tunable and correlates with several measures of search difficulty. We discuss why these properties make NM landscapes preferable to both NK landscapes and Walsh polynomials as benchmark landscape models with tunable epistasis.

A kind of epistasis‐tunable test functions for genetic algorithms

“…FDC is one of the most well known predictive measures of problem difficulty for GAs 6,28,33,34 . Jones and Forrest pointed out that, the higher FDC was, the easier a fitness function for a GA would be; they also empirically determined the intervals associated with different difficulty classes of problems, and suggested that, for a binary GA, a problem was difficult when FDC fell within ( −0.15,0.15) and was easy when FDC was smaller than or equal to −0.15 33…”

“…In general, there have been two major ways to tune the epistasic degree of a test function. 1)Coordinate rotation 1,18 : A rotation of the coordinate system can change the interdependent degree between the variables; it can turn a separable function into a partially separable function, and further into a fully non‐separable function. 2)Parameter tuning: Some simulated landscapes widely used for evaluating search strategies are with tunable epistatic interactions, eg, NK landscapes 25,26 produced within the framework of Walsh polynomials, 27 and NM problems 28 proposed within the framework of general variable interaction models 29 . The epistatic degree of these landscapes can be tuned via change to a handful of model parameters. …”

Multi-objective NM-Landscapes

“…A fitness landscape F can be defined for N variables using a general parametric interaction model of the form [2]:…”

“…, N }, where U k is a set of indices of the variables in the kth term, and the length |U k | is the order of the interaction. By convention [2], it is assumed that when U k = ∅, j∈U k xj ≡ 1. Also Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page.…”

“…by convention, we assume the model is defined for binary variables represented as xi ∈ {−1, 1}. The NM models [2] comprise the set of all general interactions models specified by Equation 1, with the following constraints: 1) All coefficients βU k are non-negative. 2) Each feature value xi ranges from negative to positive values.…”