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L. Darrell Whitley

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Journal Articles

Publisher: Journals Gateway

*Evolutionary Computation*(2016) 24 (3): 491–519.

Published: 01 September 2016

Abstract

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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 time. Bounded separable problems are also solved in time. As a result, Gray Box Optimization is able to solve many commonly used problems from the evolutional computation literature in 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. 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 time. Bounded separable problems are also solved in time. As a result, Gray Box Optimization is able to solve many commonly used problems from the evolutional computation literature in 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. 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 time. Bounded separable problems are also solved in time. As a result, Gray Box Optimization is able to solve many commonly used problems from the evolutional computation literature in 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.

Journal Articles

Publisher: Journals Gateway

*Evolutionary Computation*(2015) 23 (2): 217–248.

Published: 01 June 2015

Abstract

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Bit-flip mutation is a common mutation operator for evolutionary algorithms applied to optimize functions over binary strings. In this paper, we develop results from the theory of landscapes and Krawtchouk polynomials to exactly compute the probability distribution of fitness values of a binary string undergoing uniform bit-flip mutation. We prove that this probability distribution can be expressed as a polynomial in p , the probability of flipping each bit. We analyze these polynomials and provide closed-form expressions for an easy linear problem (Onemax), and an NP-hard problem, MAX-SAT. We also discuss a connection of the results with runtime analysis.

Journal Articles

Publisher: Journals Gateway

*Evolutionary Computation*(2013) 21 (4): 561–590.

Published: 01 November 2013

Abstract

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The frequency distribution of a fitness function over regions of its domain is an important quantity for understanding the behavior of algorithms that employ randomized sampling to search the function. In general, exactly characterizing this distribution is at least as hard as the search problem, since the solutions typically live in the tails of the distribution. However, in some cases it is possible to efficiently retrieve a collection of quantities (called moments ) that describe the distribution. In this paper, we consider functions of bounded epistasis that are defined over length- n strings from a finite alphabet of cardinality q . Many problems in combinatorial optimization can be specified as search problems over functions of this type. Employing Fourier analysis of functions over finite groups, we derive an efficient method for computing the exact moments of the frequency distribution of fitness functions over Hamming regions of the q -ary hypercube. We then use this approach to derive equations that describe the expected fitness of the offspring of any point undergoing uniform mutation. The results we present provide insight into the statistical structure of the fitness function for a number of combinatorial problems. For the graph coloring problem, we apply our results to efficiently compute the average number of constraint violations that lie within a certain number of steps of any coloring. We derive an expression for the mutation rate that maximizes the expected fitness of an offspring at each fitness level. We also apply the results to the slightly more complex frequency assignment problem, a relevant application in the domain of the telecommunications industry. As with the graph coloring problem, we provide formulas for the average value of the fitness function in Hamming regions around a solution and the expectation-optimal mutation rate.

Journal Articles

Publisher: Journals Gateway

*Evolutionary Computation*(2011) 19 (4): 597–637.

Published: 01 December 2011

Abstract

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A small number of combinatorial optimization problems have search spaces that correspond to elementary landscapes, where the objective function f is an eigenfunction of the Laplacian that describes the neighborhood structure of the search space. Many problems are not elementary; however, the objective function of a combinatorial optimization problem can always be expressed as a superposition of multiple elementary landscapes if the underlying neighborhood used is symmetric. This paper presents theoretical results that provide the foundation for algebraic methods that can be used to decompose the objective function of an arbitrary combinatorial optimization problem into a sum of subfunctions, where each subfunction is an elementary landscape. Many steps of this process can be automated, and indeed a software tool could be developed that assists the researcher in finding a landscape decomposition. This methodology is then used to show that the subset sum problem is a superposition of two elementary landscapes, and to show that the quadratic assignment problem is a superposition of three elementary landscapes.

Journal Articles

Publisher: Journals Gateway

*Evolutionary Computation*(1994) 2 (3): 249–278.

Published: 01 September 1994

Abstract

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Delta coding is an iterative genetic search strategy that dynamically changes the representation of the search space in an attempt to exploit different problem representations. Delta coding sustains search by reinitializing the population at each iteration of search. This helps to avoid the asymptotic performance typically observed in genetic search as the population becomes more homogeneous. Here, the optimization ability of delta coding is empirically compared against CHC, ESGA, GENITOR, and random mutation hill-climbing (RMHC) on a suite of well-known test functions with and without Gray coding. Issues concerning the effects of Gray coding on these test functions are addressed.