Time Complexity and Zeros of the Hypervolume Indicator Gradient Field

In multi-objective optimization the hypervolume indicator is a measure for the size of the space within a reference set that is dominated by a set of μ points. It is a common performance indicator for judging the quality of Pareto front approximations. As it does not require a-priori knowledge of the Pareto front it can also be used in a straightforward manner for guiding the search for finite approximations to the Pareto front in multi-objective optimization algorithm design.

[1]  C. Hillermeier.  Generalized Homotopy Approach to Multiobjective Optimization , 2001 .

[2]  Martin Brown,et al.  Effective Use of Directional Information in Multi-objective Evolutionary Computation , 2003, GECCO.

[3]  Anne Auger,et al.  Theory of the hypervolume indicator: optimal μ-distributions and the choice of the reference point , 2009, FOGA '09.

[4]  A. L. Custódio,et al.  Recent Developments in Derivative-Free Multiobjective Optimisation , 2012, Computational Technology Reviews.

[5]  Anne Auger,et al.  Hypervolume-based multiobjective optimization: Theoretical foundations and practical implications , 2012, Theor. Comput. Sci..

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

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

[8]  Michael Emmerich,et al.  Logarithmic-Time Updates in SMS-EMOA and Hypervolume-Based Archiving , 2013 .

[9]  Karl Bringmann,et al.  Bringing Order to Special Cases of Klee's Measure Problem , 2013, MFCS.

[10]  Thomas Bäck,et al.  Parallel Problem Solving from Nature — PPSN V , 1998, Lecture Notes in Computer Science.

[11]  Jörg Fliege,et al.  Steepest descent methods for multicriteria optimization , 2000, Math. Methods Oper. Res..

[12]  Subhash Suri,et al.  On Klee's measure problem for grounded boxes , 2012, SoCG '12.

[13]  Christian Blum,et al.  Hybrid Metaheuristics , 2010, Artificial Intelligence: Foundations, Theory, and Algorithms.

[14]  Pascal Bouvry,et al.  EVOLVE - A Bridge between Probability, Set Oriented Numerics and Evolutionary Computation , 2013, EVOLVE.

[15]  Kalyanmoy Deb,et al.  Practical Approaches to Multi-Objective Optimization , 2005 .

[16]  Kalyanmoy Deb,et al.  04461 Summary - Practical Approaches to Multi-Criterion Optimization , 2005, Practical Approaches to Multi-Objective Optimization.

[17]  Nicola Beume,et al.  Design and Management of Complex Technical Processes and Systems by means of Computational Intelligence Methods Gradient-based / Evolutionary Relay Hybrid for Computing Pareto Front Approximations Maximizing the S-Metric , 2007 .

[18]  S. Schäffler,et al.  Stochastic Method for the Solution of Unconstrained Vector Optimization Problems , 2002 .

[19]  Nicola Beume,et al.  Hypervolume based metaheuristics for multiobjective optimization , 2012 .

[20]  Carlos M. Fonseca,et al.  A Fast Dimension-Sweep Algorithm for the Hypervolume Indicator in Four Dimensions , 2012, CCCG.

[21]  Carlos A. Coello Coello,et al.  On Gradient-Based Local Search to Hybridize Multi-objective Evolutionary Algorithms , 2013, EVOLVE.

[22]  Carlos M. Fonseca,et al.  On the Computation of the Empirical Attainment Function , 2011, EMO.

[23]  Nicola Beume,et al.  On the Complexity of Computing the Hypervolume Indicator , 2009, IEEE Transactions on Evolutionary Computation.

[24]  Julian F. Miller,et al.  Genetic and Evolutionary Computation — GECCO 2003 , 2003, Lecture Notes in Computer Science.

[25]  Oliver Schütze,et al.  On Continuation Methods for the Numerical Treatment of Multi-Objective Optimization Problems , 2005, Practical Approaches to Multi-Objective Optimization.

[26]  Kalyanmoy Deb,et al.  Comparing Classical Generating Methods with an Evolutionary Multi-objective Optimization Method , 2005, EMO.

[27]  Peter A. N. Bosman,et al.  Exploiting gradient information in numerical multi--objective evolutionary optimization , 2005, GECCO '05.

[28]  Ricardo A. Baeza-Yates,et al.  A Fast Set Intersection Algorithm for Sorted Sequences , 2004, CPM.

[29]  Carlos M. Fonseca,et al.  Computing Hypervolume Contributions in Low Dimensions: Asymptotically Optimal Algorithm and Complexity Results , 2011, EMO.

[30]  Ricardo H. C. Takahashi.  Evolutionary multi-criterion optimization : 6th international conference, EMO 2011, Ouro Preto, Brazil, April 5-8, 2011 : proceedings , 2011 .

[31]  Marco Laumanns,et al.  Performance assessment of multiobjective optimizers: an analysis and review , 2003, IEEE Trans. Evol. Comput..