Abstract. This article describes a set function that maps a set of Pareto optimal points to a scalar. A proof is presented that shows that the maximization of this scalar value constitutes the necessary and sufficient condition for the function's arguments to be maximally diverse Pareto optimal solutions of a discrete, multi-objective, optimization problem. This scalar quantity, a hypervolume based on a Lebesgue measure, is therefore the best metric to assess the quality of multiobjective optimization algorithms. Moreover, it can be used as the objective function in simulated annealing (SA) to induce convergence in probability to the Pareto optima. An efficient algorithm for calculating this scalar and analysis of its complexity is presented.
Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Abstract. This article describes a set function that maps a set of Pareto optimal points to a scalar. A proof is presented that shows that the maximization of this scalar value constitutes the necessary and sufficient condition for the function's arguments to be maximally diverse Pareto optimal solutions of a discrete, multi-objective, optimization problem. This scalar quantity, a hypervolume based on a Lebesgue measure, is therefore the best metric to assess the quality of multiobjective optimization algorithms. Moreover, it can be used as the objective function in simulated annealing (SA) to induce convergence in probability to the Pareto optima. An efficient algorithm for calculating this scalar and analysis of its complexity is presented.
Abstract-Swarm Intelligence (SI) is a relatively new paradigm being applied in a host of research settings to improve the management and control of large numbers of interacting entities such as communication, computer and sensor networks, satellite constellations and more. Attempts to take advantage of this paradigm and mimic the behavior of insect swarms however often lead to many different implementations of SI. The rather vague notions of what constitutes self-organized behavior lead to rather ad hoc approaches that make it difficult to ascertain just what SI is, assess its true potential and more fully take advantage of it. This article provides a set of general principles for SI research and development. A precise definition of self-organized behavior is described and provides the basis for a more axiomatic and logical approach to research and development as opposed to the more prevalent ad hoc approach in using SI concepts.The concept of Pareto optimality is utilized to capture the notions of efficiency and adaptability. A new concept, Scale Invariant Pareto Optimality is described and entails symmetry relationships and scale invariance where Pareto optimality is preserved under changes in system states. This provides a mathematical way to describe efficient tradeoffs of efficiency between different scales and further, mathematically captures the notion of the graceful degradation of performance so often sought in complex systems.
This article presents an empirical approach that demonstrates a theoretical connection between (information theoretic) entropy measures and the finite-time performance of the simulated annealing algorithm. The methodology developed leads to several computational approaches for creating problem instances useful in testing and demonstrating the entropy/performance connection: use of generic configuration spaces, polynomial transformations between NP-hard problems, and modification of penalty parameters. In particular, the computational results show that higher entropy measures are associated with superior finite-time performance of the simulated annealing algorithm.
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