Zigzag Search for Continuous Multiobjective Optimization

A new method is proposed using a gradient-based zigzag search approach for multiobjective optimization MOO or vector optimization problems. The key idea of this method is searching around the Pareto front by applying an efficient local search procedure using the gradients of the objective functions. This local search zigzags along the Pareto surface guided by the gradients and iteratively returns the visited Pareto optima. Many continuous MOO problems have smooth objective functions and the set of the nondominated objective function values forms a regular surface in the image space. This fact motivates developing the zigzag search method for such relatively well-posed MOO problems. A simple implementation of this method, z-algorithm, is presented particularly for continuous bi-objective optimization BOO problems with well-connected Pareto optimal solutions. The efficiency of the z-algorithm is studied with a set of BOO problems and the algorithm performances are compared to those of a recently developed MOO algorithm, Pareto front approximation with adaptive weighted sum method.

[1]  Marco Laumanns,et al.  SPEA2: Improving the strength pareto evolutionary algorithm , 2001 .

[2]  Louis J. Durlofsky,et al.  Optimal Well Placement Under Uncertainty Using a Retrospective Optimization Framework , 2012 .

[3]  Chang-Hwan Im,et al.  Multimodal function optimization based on particle swarm optimization , 2006, IEEE Transactions on Magnetics.

[4]  Jing J. Liang,et al.  Comprehensive learning particle swarm optimizer for global optimization of multimodal functions , 2006, IEEE Transactions on Evolutionary Computation.

[5]  Carlos A. Coello Coello,et al.  A painless gradient-assisted multi-objective memetic mechanism for solving continuous bi-objective optimization problems , 2010, IEEE Congress on Evolutionary Computation.

[6]  G. Gary Wang,et al.  An Efficient Pareto Set Identification Approach for Multiobjective Optimization on Black-Box Functions , 2005 .

[7]  Mohit Singh,et al.  Iterative Rounding for Multi-Objective Optimization Problems , 2009, ESA.

[8]  H. Markowitz.  Portfolio Selection: Efficient Diversification of Investments , 1971 .

[9]  Kalyanmoy Deb,et al.  Multi-objective Genetic Algorithms: Problem Difficulties and Construction of Test Problems , 1999, Evolutionary Computation.

[10]  A. Stuart,et al.  Portfolio Selection: Efficient Diversification of Investments , 1959 .

[11]  Charles Audet,et al.  Mesh Adaptive Direct Search Algorithms for Constrained Optimization , 2006, SIAM J. Optim..

[12]  Hajime Kita,et al.  Multi-objective optimization by genetic algorithms: a review , 1996, Proceedings of IEEE International Conference on Evolutionary Computation.

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

[14]  Maria Cristina Recchioni,et al.  A path following method for box-constrained multiobjective optimization with applications to goal programming problems , 2003, Math. Methods Oper. Res..

[15]  K. Feldman.  Portfolio Selection, Efficient Diversification of Investments . By Harry M. Markowitz (Basil Blackwell, 1991) £25.00 , 1992 .

[16]  Panos M. Pardalos,et al.  Handbook of Multicriteria Analysis , 2010 .

[17]  P. Pardalos,et al.  Pareto optimality, game theory and equilibria , 2008 .

[18]  T. Simpson,et al.  Efficient Pareto Frontier Exploration using Surrogate Approximations , 2000 .

[19]  Amos Fiat,et al.  Algorithms - ESA 2009 , 2009, Lecture Notes in Computer Science.

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

[21]  Jong-hyun Ryu,et al.  Pareto front approximation with adaptive weighted sum method in multiobjective simulation optimization , 2009, Proceedings of the 2009 Winter Simulation Conference (WSC).

[22]  Ivo F. Sbalzariniy,et al.  Multiobjective optimization using evolutionary algorithms , 2000 .

[23]  Honggang Wang,et al.  Use of retrospective optimization for placement of oil wells under uncertainty , 2010, Proceedings of the 2010 Winter Simulation Conference.

[24]  John E. Dennis,et al.  Normal-Boundary Intersection: A New Method for Generating the Pareto Surface in Nonlinear Multicriteria Optimization Problems , 1998, SIAM J. Optim..

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

[26]  Dimitri P. Bertsekas,et al.  Nonlinear Programming , 1997 .

[27]  John E. Renaud,et al.  An interactive multiobjective optimization design strategy for decision based multidisciplinary design , 2002 .

[28]  Carlos A. Coello Coello,et al.  HCS: A New Local Search Strategy for Memetic Multiobjective Evolutionary Algorithms , 2010, IEEE Transactions on Evolutionary Computation.

[29]  Lakhmi C. Jain,et al.  Evolutionary Multiobjective Optimization , 2005, Evolutionary Multiobjective Optimization.

[30]  Charles Audet,et al.  Multiobjective Optimization Through a Series of Single-Objective Formulations , 2008, SIAM J. Optim..

[31]  H. Fawcett.  Manual of Political Economy , 1995 .

[32]  Jing J. Liang,et al.  Dynamic Multi-Swarm Particle Swarm Optimizer with a Novel Constraint-Handling Mechanism , 2006, 2006 IEEE International Conference on Evolutionary Computation.

[33]  H. Ishibuchi,et al.  Multi-objective genetic algorithm and its applications to flowshop scheduling , 1996 .

[34]  Panos M. Pardalos,et al.  A survey of recent developments in multiobjective optimization , 2007, Ann. Oper. Res..

[35]  Jasbir S. Arora,et al.  Survey of multi-objective optimization methods for engineering , 2004 .