Multiobjective Landscape Analysis and the Generalized Assignment Problem

The importance of tuning a search algorithm for the specific features of the target search space has been known for quite some time. However, when dealing with multiobjective problems, there are several twists on the conventional notions of fitness landscapes. Multiobjective optimization problems provide additional difficulties for those seeking to study the properties of the search space. However, the requirement of finding multiple candidate solutions to the problem also introduces new potentially exploitable structure. This paper provides a somewhat high-level overview of multiobjective search space and fitness landscape analysis and examines the impact of these features on the multiobjective generalized assignment problem. 1 Why Landscape Analysis One of the foremost questions facing designers of metaheuristic algorithms for any sort of problem is how the structure of the objective function will affect the behavior of the search algorithm. It is known, and quite intuitive, that incorporating problem-specific knowledge into a search algorithm can often substantially increase the performance of the algorithm. However, given multiple conflicting options for building such algorithms, comparatively little is known concerning the right choices, or even the right information necessary to make good decisions. This work examines a set of tools developed to help gain insights into how various algorithms navigate complex multiobjective search spaces. Many of these tools have previously been described in relation to conventional optimization problems. In such cases, the implications of extending the tools into the multiobjective realm are carefully examined. In this work, we propose some methods by which such relevant information may be obtained and exploited, and we apply these methods to a pair of classes of assignment problems exhibiting markedly different types of structure. 2 Fitness Landscape Analysis As more and more researchers have turned their attention to modeling search algorithm performance, a number of techniques have been proposed to classify fitness landscapes, generally corresponding to fundamental properties of a given search space. While most commonly defined in terms of classical single objective optimization, many of these properties have straightforward generalizations to the multiobjective domain. However, multiobjective optimization introduces additional features which may be analyzed and exploited by search algorithms. By studying these features, one should be able to design metaheuristic algorithms to better take advantage of the peculiarities of multiobjective optimization of a given problem. Often, multiobjective search algorithms are defined in terms of a simpler, single objective algorithm. The n objectives of the original problem are scalarized using a particular weight vector, and the resulting single objective problem is attacked using the component search method. In these cases, the multiobjective problem may be completely characterized, for the purpose of modeling the performance of such an algorithm, by a family of related fitness landscapes. Each landscape is a window on the problem as viewed through a specific weight vector. Abstractly, one may thus consider the ruggedness of adjacent landscapes. In fact, this notion of the similarity between nearby landscapes is what determines in part the success of different types of multiobjective local search algorithms. Following convention established by analysis of single objective landscapes, we may characterize a multiobjective problem as “smooth” if small changes to the underlying weight vector impose small changes on the fitness landscape. Conversely, a “rugged” multiobjective problem is one in which making a small change to the weight vector drastically alters the resulting fitness landscape.ly, one may thus consider the ruggedness of adjacent landscapes. In fact, this notion of the similarity between nearby landscapes is what determines in part the success of different types of multiobjective local search algorithms. Following convention established by analysis of single objective landscapes, we may characterize a multiobjective problem as “smooth” if small changes to the underlying weight vector impose small changes on the fitness landscape. Conversely, a “rugged” multiobjective problem is one in which making a small change to the weight vector drastically alters the resulting fitness landscape. Given a single good solution to a multiobjective optimization problem, the difficulty in finding other good solutions is largely determined by the smoothness of the family of landscapes. It is somewhat intuitive that smoothness implies that a good solution on a particular landscape should be nearby to good solutions on nearby landscapes. This spatial locality makes algorithms which attempt to build from one solution to a multiobjective problem to find many others more attractive. On the other hand, as the family of landscapes becomes more rugged, the information gained by finding one good solution becomes less useful in finding others. The following sections describe a number of potentially useful metrics by which fitness landscapes, both single and multiobjective, may be characterized. In general, many of the tools in common use for analysis of single objective landscapes have fairly straightforward generalizations to the multiobjective realm. In addition, multiobjective algorithms which consist entirely of a sequence of independent runs of some underlying single objective optimization method may be directly studied using the single objective tools. However, it is also true that multiobjective optimization provides additional opportunities to exploit problem knowledge, and much of the goal of this work is to study these opportunities and apply the resulting knowledge to the problem of designing more effective algorithms. The remainder of this chapter is focused on defining a number of tools by which we may obtain such useful information about a given multiobjective problem instance. 2.1 Distribution of Local and Pareto Optima Intuitively, the number and distribution of local optima would seem to have a profound impact on the performance of a general purpose search algorithm. A problem with but a single local optimum is by definition unimodal, and thus easily solved by any number of simple algorithms. As the number of local optima increase, the chances of becoming trapped in a local optimum are increased correspondingly. However, the distribution of local optima throughout the space is at least as important as the number of such optima. Classical notions such as deception are rooted entirely in the notion of unfortunate distributions of local optima. One of the best-known examples of real-world problems with very different optima distributions is the comparison between the traveling salesman problem (TSP) and the quadratic assignment problem (QAP). In the TSP, problem instances exhibit what is commonly known as a “Big Valley” structure [1]. This term refers to the phenomena that almost all local optima are concentrated around a line that approaches the global optima, with the tour lengths of points along that line tending to increase as the distance to the global optimum increases. In contrast, QAP instances tend to exhibit almost no structure when viewed in this same manner. The local optima for a typical QAP instance are very nearly uniformly distributed throughout the search space, with many, perhaps most, of all local optima lying close to the maximum possible distance from the global optimum. The difference in performance of a local search algorithm on TSP versus QAP is therefore quite dramatic. From a multiobjective standpoint, many Pareto optimal solutions are also global optima of some single objective problem. Most commonly, given a Pareto front which is globally convex, there exists a weight vector for each Pareto optimal solution, a scalarization of the problem by which would result in the solution being the globally optimal solution of the resulting single objective problem. Thus, the distribution of local optima affects multiobjective problems just as it does for their single objective counterparts. However, multiobjective landscapes add an additional consideration, in that different Pareto optima are not generally local optima of the same single objective slice of the landscape. Therefore, the distribution of Pareto optima is in some sense a different aspect of the landscape than is the distribution of local optima leading to a single point on the Pareto front. In essense, you have to find both solutions, and the entire landscape can and often does change underneath you as you try to switch from one to the other. 2.2 Fitness Distance Correlation Fitness distance correlation as a tool for modeling algorithm performance is based on the notion that good local optima should be near to the global optimum in terms of fitness as well. If this is the case, in principle there should be a clear trail from any local optimum to the global optimum in which each step requires only small changes to the current solution. If instead, large jumps are required to move from a local optimum to a better solution nearer to the global optimum, most search algorithms may be expected to suffer. Primarily defined in terms of single-objective optimization, fitness distance correlation is the correlation coefficient between the distance in objective space and the distance in parameter space between a set of randomly distributed local optima and the respective nearest global optimum to each. The standard Pearson correlation coefficient may be used to describe the results, although some useful information is not captured by this single summary statistic. Instead, scatter plots of parameter space distance versus objective space distance are often reported. In the context of multiobjective optimization, the basic distinction is that the set of global optima is taken to be the set of nondominated solutions. While conceptually a simple extension, each Pareto optimal solution may or may not be the optimum of some mono-objective problem associated with a particular set of weights. The novelty in such a formulation is that, because each solution is, in essence, the optimum of a different fitness function, the correlations between nondominated solutions need bear no resemblance to the correlation between different local optima of a single function. Thus, considering the correlation between nondominated solutions can provide very useful information concerning the relative difficulty of moving “along” the Pareto front. There are possibly other, more useful, ways to generalize this concept to multiobjective landscapes. The basic restriction of FDC is that one needs to get a single number indicating the distance to each optima. While one can certainly treat the Euclidean distance between a fitness vectors as the requisite metric, one may also consider, for example, the angle between vectors. If we treat distance as being defined by the angle between fitness vectors, then the nearest optimum will be, in a sense, “aligned” with the solution, in that they will have their component fitness values in the nearest proportion with one another. The nearest Pareto optimum under this definition will be that which lies adjacent on the Pareto front. The impact of these different choices on the resulting analysis is still an open question, but it is worthwhile to keep in mind the various possibilities that arise when dealing with vector valued optimization.