An Optimality Theory Based Proximity Measure for Evolutionary Multi-Objective and Many-Objective Optimization

Evolutionary multi- and many-objective optimization (EMO) methods attempt to find a set of Pareto-optimal solutions, instead of a single optimal solution. To evaluate these algorithms, performance metrics either require the knowledge of the true Pareto-optimal solutions or, are ad-hoc and heuristic based. In this paper, we suggest a KKT proximity measure (KKTPM) that can provide an estimate of the proximity of a set of trade-off solutions from the true Pareto-optimal solutions. Besides theoretical results, the proposed KKT proximity measure is computed for iteration-wise trade-off solutions obtained from specific EMO algorithms on two, three, five and 10-objective optimization problems. Results amply indicate the usefulness of the proposed KKTPM as a termination criterion for an EMO algorithm.

[1]  P. S. Manoharan,et al.  Evolutionary algorithm solution and KKT based optimality verification to multi-area economic dispatch , 2009 .

[2]  Kalyanmoy Deb,et al.  An Evolutionary Many-Objective Optimization Algorithm Using Reference-Point-Based Nondominated Sorting Approach, Part I: Solving Problems With Box Constraints , 2014, IEEE Transactions on Evolutionary Computation.

[3]  Kalyanmoy Deb,et al.  Finding trade-off solutions close to KKT points using evolutionary multi-objective optimization , 2007, 2007 IEEE Congress on Evolutionary Computation.

[4]  C. R. Bector,et al.  Principles of Optimization Theory , 2005 .

[5]  Kaisa Miettinen,et al.  Nonlinear multiobjective optimization , 1998, International series in operations research and management science.

[6]  Kalyanmoy Deb,et al.  Faster Hypervolume-Based Search Using Monte Carlo Sampling , 2008, MCDM.

[7]  Kalyanmoy Deb,et al.  Approximate KKT points and a proximity measure for termination , 2013, J. Glob. Optim..

[8]  Andrzej P. Wierzbicki,et al.  The Use of Reference Objectives in Multiobjective Optimization , 1979 .

[9]  Lothar Thiele,et al.  Comparison of Multiobjective Evolutionary Algorithms: Empirical Results , 2000, Evolutionary Computation.

[10]  Kalyanmoy Deb,et al.  Multi-objective optimization using evolutionary algorithms , 2001, Wiley-Interscience series in systems and optimization.

[11]  Kalyanmoy Deb,et al.  An Optimality Theory-Based Proximity Measure for Set-Based Multiobjective Optimization , 2016, IEEE Transactions on Evolutionary Computation.

[12]  Simon French,et al.  Multiple Criteria Decision Making: Theory and Application , 1981 .

[13]  J. M. Martínez,et al.  On sequential optimality conditions for smooth constrained optimization , 2011 .

[14]  Matthias Ehrgott,et al.  Multicriteria Optimization , 2005 .

[15]  Kalyanmoy Deb,et al.  Investigating EA solutions for approximate KKT conditions in smooth problems , 2010, GECCO '10.

[16]  Kalyanmoy Deb,et al.  A fast and elitist multiobjective genetic algorithm: NSGA-II , 2002, IEEE Trans. Evol. Comput..

[17]  Kalyanmoy Deb,et al.  Scope of stationary multi-objective evolutionary optimization: a case study on a hydro-thermal power dispatch problem , 2008, J. Glob. Optim..

[18]  R. Lyndon While,et al.  A faster algorithm for calculating hypervolume , 2006, IEEE Transactions on Evolutionary Computation.