The measure of pareto optima: Applications to multi-objective metaheuristics

This article describes a set function that maps a set of Pareto optimal points to a scalar. A theorem 1 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, polynomial-time algorithm for calculating this scalar and an analysis of its complexity is also presented.

[1]  Lothar Thiele,et al.  Multiobjective Optimization Using Evolutionary Algorithms - A Comparative Case Study , 1998, PPSN.

[2]  Susan H. Rodger,et al.  A new approach to the dynamic maintenance of maximal points in a plane , 1990, Discret. Comput. Geom..

[3]  Piotr Czyzżak,et al.  Pareto simulated annealing—a metaheuristic technique for multiple‐objective combinatorial optimization , 1998 .

[4]  Eckart Zitzler,et al.  Evolutionary algorithms for multiobjective optimization: methods and applications , 1999 .

[5]  F. B. Hildebrand,et al.  Introduction To Numerical Analysis , 1957 .

[6]  David W. Corne,et al.  Approximating the Nondominated Front Using the Pareto Archived Evolution Strategy , 2000, Evolutionary Computation.

[7]  Peter J. Fleming,et al.  On the Performance Assessment and Comparison of Stochastic Multiobjective Optimizers , 1996, PPSN.

[8]  Sanjiv Kapoor.  Dynamic Maintenance of Maxima of 2-d Point Sets , 2000, SIAM J. Comput..

[9]  D. Mitra,et al.  Convergence and finite-time behavior of simulated annealing , 1985, 1985 24th IEEE Conference on Decision and Control.

[10]  H. Schwefel,et al.  Approximating the Pareto Set: Concepts, Diversity Issues, and Performance Assessment , 1999 .

[11]  Shapour Azarm,et al.  Metrics for Quality Assessment of a Multiobjective Design Optimization Solution Set , 2001 .

[12]  M. Fleischer.  Generalized Cybernetic Optimization: Solving Continuous Variable Problems , 1999 .

[13]  Gary B. Lamont,et al.  Multiobjective Evolutionary Algorithms: Analyzing the State-of-the-Art , 2000, Evolutionary Computation.

[14]  David Williams,et al.  Probability with Martingales , 1991, Cambridge mathematical textbooks.

[15]  C. Reeves.  Modern heuristic techniques for combinatorial problems , 1993 .