Mapping the global structure of TSP fitness landscapes

The global structure of combinatorial landscapes is not fully understood, yet it is known to impact the performance of heuristic search methods. We use a so-called local optima network model to characterise and visualise the global structure of travelling salesperson fitness landscapes of different classes, including random and structured real-world instances of realistic size. Our study brings rigour to the characterisation of so-called funnels, and proposes an intensive and effective sampling procedure for extracting the networks. We propose enhanced visualisation techniques, including 3D plots and the incorporation of colour, sizes and widths, to reflect relevant aspects of the search process. This brings an almost tangible new perspective to the landscape and funnel metaphors. Our results reveal a much richer global structure than the suggestion of a ‘big-valley’ structure. Most landscapes of the tested instances have multiple valleys or funnels; and the number, disposition and interaction of these funnels seem to relate to search difficulty on the studied landscapes. We also find that the structured TSP instances feature high levels of neutrality, an observation not previously reported in the literature. We then propose ways of analysing and visualising these neutral landscapes.     

Searching with cooperators

Biological evolution serves as a blueprint for the design of search and optimization algorithms, and has generated vast number of research prototypes as well as industrial implementations since it was first proposed to solve complex engineering problems the 1960s [1], [2], [3]. Underlying this is the idea that the evolutionary forces of mutation, recombination, reproduction and selection can drive the population towards better adapted solution over time, effectively solving an optimization problem by navigating the fitness landscape they inhabit. A much overlooked evolutionary force in the design of better algorithms so far has been cooperation, shown to be crucial to shape individual and group behavior at multiple scales of interaction. Here we explore the ability of cooperative dynamics to further increase the efficiency of evolutionary strategies. For this, we perform computer experiments on a variety of landscapes of increasing complexity. We present evidence suggesting that cooperative dynamics are able to naturally balance exploration and exploitation of local maxima, via endless cycles of cooperation (where local maxima are exploited) and defection (where new areas are explored). Cooperative strategies prove to be more robust to landscape ruggedness than evolutionary strategies which never cooperate, always cooperate, or cooperate randomly. Furthermore, our simulations show that the cooperative dynamics are invariant to the complexity of the landscape, hinting at the possibility that cooperation strategies may be able to absorb and exploit local information to keep the exploration–exploration tradeoff invariant across a range of environments.     

Fitness landscape analysis for the no-wait flow-shop scheduling problem

The fitness landscape of the no-wait (continuous) flow-shop scheduling problem is investigated by examining the ruggedness of the landscape and the correlation between the quality of a solution and its distance to an optimal solution. The results confirm the presence of a big valley structure as known from other combinatorial optimization problems. The suitability of the landscape for search with evolutionary computation and local search methods is discussed. The observations are validated by experiments with two evolutionary algorithms.     

IMPACT OF RISK AVERSION ON THE OPTIMAL ROTATION WITH STOCHASTIC PRICE

Abstract This paper examines the effect of risk aversion on the optimal rotation when the stumpage price is stochastic. Assuming that the stumpage price is normally distributed, we show that the optimal rotation under risk aversion may be shorter than, equal to, or longer than the corresponding optimal rotation under risk neutrality. Which of these cases holds true depends on the interest rate and the real regeneration cost, and can be determined based on the marginal variance (i.e., the derivative of the variance function with respect to rotation age) evaluated at the optimal rotation under risk neutrality. Furthermore, we show that there exists a monotone continuous curve, which divides the interest rate‐regeneration cost space into two regions where risk aversion affects the optimal rotation differently. For a given interest rate, risk aversion shortens (prolongs) the optimal rotation if the regeneration cost lies below (above) the curve. Along the separating curve the optimal rotation under risk aversion coincides with the optimal rotation under risk neutrality. Two examples are presented to demonstrate the separating curve and the impacts of risk aversion on the optimal rotation.     

Robust and neutral methods for aggregating preferences into an outranking relation

Definitions are proposed for the concepts of robustness and neutrality of a decision-aid method. They are illustrated on the problem of aggregating preferences into an outranking relation in the presence of an importance relation on the criteria. We show that the concepts allow the analyst to better understand how the methods are grounded and provide him with arguments to justify the choice of one method instead of another.     

THE TRANSFORMATION OF LANDSCAPE: MODELING POLICY AND SOCIAL IMPACTS ON THE AGRICULTURAL LANDSCAPE OF LESVOS

ABSTRACT. The term landscape is more and more used as an “umbrella‘ concept, covering a series of cultural, productive and ecological processes. In order to uncover mechanisms, monitor transformations and predict changes, a complicated set of interacting factors has to be taken into account. This paper presents a model for estimating social and policy impacts on agricultural landscapes, based on the assumption that agricultural landscapes are shaped at” macro “(landscape) level by” micro “interventions at farm level. The model consists of three parts: an” ecological processes “part, which deals with processes that shape the ecological and aesthetic value of a landscape, a” population dynamics “part, which examines farmer population dynamics and a” policy impact “part, which deals with direct or indirect impacts on farming systems and farmer dynamics and refers to CAP Rural Development Measures. The model is applied for the olive and graze land agricultural landscapes of Lesvos (Greece). Results, apart from revealing landscape change patterns; help to illustrate some mechanisms behind this change and indicate that Rural Development Measures are inherent with minor but important malfunctions that cannot lead to sustainable landscape management and rural development in the area.     

Landscapes of Investigation

According to many observations, traditional mathematics education falls within the exercise paradigm. This paradigm is contrasted with landscapes of investigation serving as invitations for students to be involved in processes of exploration and explanation. The distinction between the exercise paradigm and landscapes of investigation is combined with a distinction between three different types of reference which might provide mathematical concepts and classroom activities with meaning: references to mathematics; references to a semi-reality, and references to a real-life situation. The six possible learning milieus are illustrated by examples. Moving away from the exercise paradigm and in the direction of landscapes of investigation may help to abandon the authorities of the traditional mathematics classroom and to make students the acting subjects in their learning processes. Moving away from reference to pure mathematics and in the direction of real life references may help to provide resources for reflection on mathematics and its applications. My hope is that finding a route among the different milieus of learning may provide new resources for making the students both acting and reflecting and in this way providing mathematics education with a critical dimension.     

The theory of elementary landscapes

When joined to a stipulated neighborhood diagraph, an objective functin defined on the solution space of a real combinatorial optimization problem forms a landscape. Grover shows that landscapes satisfying a certain difference equation have properties favorable to local search.Studying only symmetric and regular neighborhood diagraphs, Stadler defines elementary landscapes as those which can be realized as an eigenvector of the Laplacian of the neighborhood diagraph, and shows that such landscapes satisfy Grover's difference equation.Recent developments in algebraic graph theory support a new definition of the graph Laplacian which we use to extend the notion of elementary landscapes to neighborhood diagraphs which may be neither regular nor symmetric. This paper uses the new definition to extend the notion of elementary landscapes so that they characterize landscapes satisfying Grover's wave equation.We extend some known results to these more general elementary landscapes and analyse the types which may occur.     

Correlation analysis of coupled fitness landscapes

The correlation structure of fitness landscapes is a much used measure to characterize and classify various types of landscapes. However, analyzing the correlation structure of fitness landscapes has so far been restricted to static landscapes only. Here, we investigate the correlation structure of coupled, or dynamic, fitness landscapes. Using the NKC model of coevolution, we apply a correlation analysis on various instances of this model and present the results. One of the main goals of this article is thus to show that a previously introduced correlation analysis can be successfully extended to coupled fitness landscapes. Furthermore, our analysis shows that this provides meaningful and interesting results that can contribute to a better understanding of coevolution in general. © 2005 Wiley Periodicals, Inc. Complexity 10: 41–49, 2005     

The Owen and Banzhaf–Owen values revisited

In this work, we consider games with coalitional structure. We afford two new parallel axiomatic characterizations for the well-known Owen and Banzhaf–Owen coalitional values. Two properties are common to both characterizations: a property of balanced contributions and a property of neutrality. The results prove that the main difference between these two coalitional values is that the former is efficient, while the latter verifies a property of 2-efficiency.     

Canonical approximation of fitness landscapes

We present a method for approximating a fitness landscape as a superposition of “elementary” landscapes. Given a correlation function of the landscape in question we show that the relative amplitudes of contributions with p-ary interactions can be computed. We show an application to RNA free energy landscapes. © 1996 John Wiley & Sons, Inc.     

On neutrality of preferences on acts with respect to use of proxy outcomes

Construing acts as functions on a set of states to the set of lotteries on ultimate outcomes, this partly expository paper cautions that, in non-Savage choice theories, the popular technique of replacing multiattributed ultimate outcomes by equally desirable and crisper proxies may misguide the choice of act, in that preferences between the induced proxy acts may not mirror preferences between their original counterparts. When this pathology cannot occur, preferences and the associated choice theory are said to beneutral as regards proxy substitution. Sufficient conditions for neutrality include Monotone Dominance, Transitivity, and a restricted form of Linearity. The Linearity property can be circumvented by direct assignment of proxy certainty equivalents to outcome lotteries rather than to ultimate outcomes; then preferences are said to besemineutral if the corresponding proxy acts mirror preferences between their original counterparts. Relationships of these problem transformations are noted and certain normative choice theories are examined as to neutrality of preference with respect to use of arbitrary proxies.     

Fast Fourier Transform for Fitness Landscapes

We cast some classes of fitness landscapes as problems of spectral analysis on various Cayley graphs. In particular, landscapes derived from RNA folding are realized on Hamming graphs and analyzed in terms of Walsh transforms; assignment problems are interpreted as functions on the symmetric group and analyzed in terms of the representation theory of S_n. We show that explicit computation of the Walsh/Fourier transforms is feasible for landscapes with up to 10⁸ configurations using fast Fourier transform techniques. We find that the cost function of a linear sum assignment problem involves only the defining representation of the symmetric group, while quadratic assignment problems are superpositions of the representations indexed by the partitions (n), (n−1,1), (n−2,2), and (n−2,1,1). These correspond to the four smallest eigenvalues of the Laplacian of the Cayley graph obtained by using transpositions as the generating set on S_n.     

Equal treatment, symmetry and Banzhaf value axiomatizations

The Banzhaf value is the only value satisfying the equal treatment, dummy player and marginal contributions conditions and neutrality of some linear operators on the spaces of games. Under some of these neutrality assumptions, equal treatment can be replaced by even weaker conditions. For linear values having the dummy player property, equal treatment is equivalent to symmetry. All these properties together imply the marginal contributions condition, but in some Banzhaf value axiomatizations marginal contributions cannot be a substitute for linearity. Received: December 1997/Revised version: May 2001     

Resembling Particulars: What Nominalism?

This paper examines a recent proposal for reviving so-called resemblance nominalism. It is argued that, although consistent, it naturally leads to trope theory upon examination for reasons having to do with the appeal of neutrality as regards certain non-trivial ontological theses.     

On the Hardness of the Quadratic Assignment Problem with Metaheuristics

Meta-heuristics are a powerful way to approximately solve hard combinatorial optimization problems. However, for a problem, the quality of results can vary considerably from one instance to another. Understanding such a behaviour is important from a theoretical point of view, but also has practical applications such as for the generation of instances during the evaluation stage of a heuristic.In this paper we propose a new complexity measure for the Quadratic Assignment Problem in the context of metaheuristics based on local search, e.g. simulated annealing. We show how the ruggedness coefficient previously introduced by the authors, in conjunction with the well known concept of dominance, provides important features of the search space explored during a local search algorithm, and gives a rather precise idea of the complexity of an instance for these heuristics. We comment previous experimental studies concerning tabu search methods and genetic algorithms with local search in the light of our complexity measure. New computational results with simulated annealing and taboo search are presented.     

Working principles,behavior, and performance of MOEAs on MNK-landscapes

This work studies the working principles, behavior, and performance of multiobjective evolutionary algorithms (MOEAs) on multiobjective epistatic fitness functions with discrete binary search spaces by using MNK-landscapes. First, we analyze the structure and some of the properties of MNK-landscapes under a multiobjective perspective by using enumeration on small landscapes. Then, we focus on the performance and behavior of MOEAs on large landscapes. We organize our study around selection, drift, mutation, and recombination, the four major and intertwined processes that drive adaptive evolution over fitness landscapes. This work clearly shows pros and cons of the main features of MOEAs, gives a valuable guide for the practitioner on how to set up his/her algorithm, enhance MOEAs, and presents useful insights on how to design more robust and efficient MOEAs.     

The Dirichlet Distribution and Process through Neutralities

A new characterization of the Dirichlet distribution, based on the notion of complete neutrality and a regression version of neutrality, is derived. It unifies earlier characterizations by James and Mosimann (Ann. Stat. 8, 183–189, 1980) and by Seshadri and Wesołowski (Sankhyā, A 65, 248–291, 2003). Also new results on identification of the Dirichlet process in the class of neutral-to-the-right processes are obtained. The proof of the main result makes an extensive use of the method of moments.     

Stochastic Programming Models for Decreasing Risk Aversion

For a stochastic programming problem with simple recourse, we show how to formulate and analyze a model that encodes the common risk attitude of decreasing risk aversion. We discuss a class of linear fractional utility functions that represent this risk attitude and show that for such a utility function the resulting nonlinear model can be reduced to a linear programming model. The linear model, moreover, has only a slight percentage increase in the number of constraints as compared with the usual linear model representing risk neutrality.