Gray Box Optimization for Mk Landscapes (NK Landscapes and MAX-kSAT)

Abstract

This article investigates Gray Box Optimization for pseudo-Boolean optimization problems composed of M subfunctions, where each subfunction accepts at most k variables. We will refer to these as Mk Landscapes. In Gray Box Optimization, the optimizer is given access to the set of M subfunctions. We prove Gray Box Optimization can efficiently compute hyperplane averages to solve non-deceptive problems in [Formula: see text] time. Bounded separable problems are also solved in [Formula: see text] time. As a result, Gray Box Optimization is able to solve many commonly used problems from the evolutional computation literature in [Formula: see text] evaluations. We also introduce a more general class of Mk Landscapes that can be solved using dynamic programming and discuss properties of these functions. For certain type of problems Gray Box Optimization makes it possible to enumerate all local optima faster than brute force methods. We also provide evidence that randomly generated test problems are far less structured than those found in real-world problems.

Optimizing one million variable NK landscapes by hybridizing deterministic recombination and local search

“…An important tool which can be constructed under Gray Box Optimization is the Variable Interaction Graph (VIG) [11]. e VIG is a graph G = (V , E), where V is the set of Boolean variables and edges E contains all the pairs of variables (x i , x j ) that have nonlinear interactions.…”

“…We will refer to variables using numbers, e.g., 9 = x 9 . e NK Landscape sums over the following 18 subfunctions: f 0 (0, 6, 14) f 5 (5, 4, 2) f 10 (10, 2, 17) f 15 (15, 7, 13) f 1 (1, 0, 6) f 6 (6, 10, 13) f 11 (11,16,17) (8,3,6) f 13 (13, 12, 15) f 4 (4, 1, 14) f 9 (9, 11, 14) f 14 (14, 4, 16) From these subfunctions, assume we extract the nonlinear interactions that are shown in Figure 1. In this example, every pair of variables that appear together in a subfunction has a nonlinear interaction.…”

“…ese functions are also called Mk Landscapes by Whitley et al [11]. Well known examples of these kind of functions are NK Landscapes (with k = K + 1), MAX-kSAT and Unconstrained adratic Optimization (with k = 2).…”

“…If f i depends on x i and other K = k − 1 random variables, the model is random. ere are some other models in between [11], but the adjacent and random models are extreme in the sense that one is very easy to solve and the other is very hard to solve. Adjacent NKQ landscapes can be optimized in polynomial time O (N ) using dynamic programming [12].…”

Enhancing partition crossover with articulation points analysis

“…In these problems, the objective function can be expressed as a sum of M subfunctions. In Gray-Box optimization [7], the optimizer is given access to these M subfunctions. The optimizer does not need to know the specific application, nevertheless useful problem structure can be extracted from the subfunctions.…”