Using convex quadratic approximation as a local search operator in evolutionary multiobjective algorithms

Local search techniques based on Convex Quadratic Approximation (CQA) of functions are studied here, in order to speed up the convergence and the quality of solutions in evolutionary multiobjective algorithms. The hybrid methods studied here pick up points from the nondominated population and determine a CQA for each objective function. Since the CQA of the functions and the respective weighted sums are convex, fast deterministic methods can be used in order to generate approximated Pareto-optimal solutions from the approximated functions. A new scheme is proposed in this paper, using a CQA model that represents a lower bound for the function points, which can be solved via linear programming. This scheme and also another one using the methodology of linear matrix inequality (LMI) for CQA are coupled with a canonical implementation of the NSGA-II. Comparison tests are performed, using Monte Carlo simulations, considering the S-metric with an equivalent final number of evaluated objective functions and the algorithm execution time. The results indicate that the proposed scheme is promising.

[1]  I. Barrodale.  L1 Approximation and the Analysis of Data , 1968 .

[2]  David Corne,et al.  The Pareto archived evolution strategy: a new baseline algorithm for Pareto multiobjective optimisation , 1999, Proceedings of the 1999 Congress on Evolutionary Computation-CEC99 (Cat. No. 99TH8406).

[3]  Roummel F. Marcia,et al.  Iterative Convex Quadratic Approximation for Global Optimization in Protein Docking , 2005, Comput. Optim. Appl..

[4]  Hisao Ishibuchi,et al.  Balance between genetic search and local search in memetic algorithms for multiobjective permutation flowshop scheduling , 2003, IEEE Trans. Evol. Comput..

[5]  David E. Goldberg,et al.  A niched Pareto genetic algorithm for multiobjective optimization , 1994, Proceedings of the First IEEE Conference on Evolutionary Computation. IEEE World Congress on Computational Intelligence.

[6]  Jos F. Sturm,et al.  A Matlab toolbox for optimization over symmetric cones , 1999 .

[7]  Pablo Moscato,et al.  On Evolution, Search, Optimization, Genetic Algorithms and Martial Arts : Towards Memetic Algorithms , 1989 .

[8]  T. J. Rivlin.  An Introduction to the Approximation of Functions , 2003 .

[9]  Stephen P. Boyd,et al.  Linear Matrix Inequalities in Systems and Control Theory , 1994 .

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

[11]  Peter J. Fleming,et al.  An Overview of Evolutionary Algorithms in Multiobjective Optimization , 1995, Evolutionary Computation.

[12]  Pablo Moscato,et al.  Memetic algorithms: a short introduction , 1999 .

[13]  K. Dill,et al.  CGU: An Algorithm for Molecular Structure Prediction , 1997 .

[14]  E. Yaz.  Linear Matrix Inequalities In System And Control Theory , 1998, Proceedings of the IEEE.

[15]  L. Goddard.  Approximation of Functions , 1965, Nature.

[16]  Kalyanmoy Deb,et al.  Muiltiobjective Optimization Using Nondominated Sorting in Genetic Algorithms , 1994, Evolutionary Computation.

[17]  Edmund K. Burke,et al.  The Design of Memetic Algorithms for Scheduling and Timetabling Problems , 2005 .

[18]  M. Powell,et al.  Approximation theory and methods , 1984 .

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

[20]  Yew-Soon Ong,et al.  A Probabilistic Memetic Framework , 2009, IEEE Transactions on Evolutionary Computation.

[21]  J. G. Hayes,et al.  Numerical Approximations to Functions and Data. , 1971 .

[22]  Pablo Moscato,et al.  A parallel memetic algorithm applied to the total tardiness machine scheduling problem , 2006, Proceedings 20th IEEE International Parallel & Distributed Processing Symposium.

[23]  L. Ghaoui,et al.  History of linear matrix inequalities in control theory , 1994, Proceedings of 1994 American Control Conference - ACC '94.

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

[25]  Joshua D. Knowles,et al.  M-PAES: a memetic algorithm for multiobjective optimization , 2000, Proceedings of the 2000 Congress on Evolutionary Computation. CEC00 (Cat. No.00TH8512).

[26]  J. G. Hayes,et al.  Numerical Approximations to Functions and Data. , 1971 .

[27]  Frederico G. Guimarães,et al.  Local Search with Quadratic Approximations into Memetic Algorithms for Optimization with Multiple Criteria , 2008, Evolutionary Computation.

[28]  Julie C. Mitchell,et al.  Coupled optimization in protein docking , 1999, RECOMB.

[29]  J. Ben Rosen,et al.  Protein Structure and Energy Landscape Dependence on Sequence Using a Continuous Energy Function , 1997, J. Comput. Biol..

[30]  Roummel F. Marcia,et al.  Convex Quadratic Approximation , 2004, Comput. Optim. Appl..

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

[32]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[33]  Lei Zhang,et al.  Accurate Solution to Overdetermined Linear Equations with Errors Using L1 Norm Minimization , 2000, Comput. Optim. Appl..

[34]  Lothar Thiele,et al.  Multiobjective evolutionary algorithms: a comparative case study and the strength Pareto approach , 1999, IEEE Trans. Evol. Comput..

[35]  Carlos Artemio Coello-Coello,et al.  Theoretical and numerical constraint-handling techniques used with evolutionary algorithms: a survey of the state of the art , 2002 .

[36]  M. Powell.  On the Lagrange functions of quadratic models that are defined by interpolation* , 2001 .