Target shape design optimization by evolving B-splines with cooperative coevolution

Graphical abstractSince real-world design optimization is often computationally expensive, target shape design optimization problems (TSDOPs) was used as miniature model to check algorithmic performance for general shape design. With B-spline and adaptive encoding as representation, the TSDOPs can be modeled as complex numerical optimization problems which can be decomposed into two subproblems. i.e. components for control points and knot vector. A new cooperative coevolution paradigm is proposed to evolve the control points and knot vector. The final designed approaches are expected to be effective for the original real-world design optimization problems. Display Omitted HighlightsA framework was proposed to evolve the two components of B-spline in a reasonable cooperated fashion. It could increase the freedom of knot vector greatly, which is desired for a better exploration in the search space.The difficulty of TSDOPs is reduced with the natural decomposition of control points and knot vector. It is also helpful for EAs exploring the search space easier in a tight time budget.An algorithm named CMA-ES-CC was formulated to evaluate the performance of the new coevolutionary paradigm.Experiments were conducted based on two target shapes. It was found that the performance of CMA-ES-CC was not only good, but also stable. The comparison with six other EAs for TSDOPs suggested that the results of CMA-ES-CC were significantly better. With high reputation in handling non-linear and multi-model problems with little prior knowledge, evolutionary algorithms (EAs) have successfully been applied to design optimization problems as robust optimizers. Since real-world design optimization is often computationally expensive, target shape design optimization problems (TSDOPs) have been frequently used as efficient miniature model to check algorithmic performance for general shape design. There are at least three important issues in developing EAs for TSDOPs, i.e., design representation, fitness evaluation and evolution paradigm. Existing work has mainly focused on the first two issues, in which (1) an adaptive encoding scheme with B-spline has been proposed as a representation, and (2) a symmetric Hausdorff distance based metric has been used as a fitness function. But for the third issue, off-the-shelf EAs were used directly to evolve B-spline control points and/or knot vector. In this paper, we first demonstrate why it is unreasonable to evolve the control points and knot vector simultaneously. And then a new coevolutionary paradigm is proposed to evolve the control points and knot vector of B-spline separately in a cooperative manner. In the new paradigm, an initial population is generated for both the control points, and the knot vector. The two populations are evolved mostly separately in a round-robin fashion, with only cooperation at the fitness evaluation phase. The new paradigm has at least two significant advantages over conventional EAs. Firstly, it provides a platform to evolve both the control points and knot vector reasonably. Secondly, it reduces the difficulty of TSDOPs by decomposing the objective vector into two smaller subcomponents (i.e., control points and knot vector). To evaluate the efficacy of the proposed coevolutionary paradigm, an algorithm named CMA-ES-CC was formulated. Experimental studies were conducted based on two target shapes. The comparison with six other EAs suggests that the proposed cooperative coevolution paradigm is very effective for TSDOPs.

[1]  César Hervás-Martínez,et al.  COVNET: a cooperative coevolutionary model for evolving artificial neural networks , 2003, IEEE Trans. Neural Networks.

[2]  Yaochu Jin,et al.  Adaptive encoding for aerodynamic shape optimization using evolution strategies , 2001, Proceedings of the 2001 Congress on Evolutionary Computation (IEEE Cat. No.01TH8546).

[3]  Xin Yao,et al.  Target shape design optimization by evolving splines , 2007, 2007 IEEE Congress on Evolutionary Computation.

[4]  K. Giannakoglou,et al.  Aerodynamic shape design using evolutionary algorithms and new gradient-assisted metamodels , 2006 .

[5]  Qiuzhen Lin,et al.  A novel hybrid multi-objective immune algorithm with adaptive differential evolution , 2015, Comput. Oper. Res..

[6]  Les A. Piegl,et al.  The NURBS Book , 1995, Monographs in Visual Communication.

[7]  Tomasz Arciszewski,et al.  Evolutionary computation and structural design: A survey of the state-of-the-art , 2005 .

[8]  Qingfu Zhang,et al.  Global path planning of wheeled robots using multi-objective memetic algorithms , 2015, Integr. Comput. Aided Eng..

[9]  Bernhard Sendhoff,et al.  Target shape design optimization with evolutionary computation , 2003, The 2003 Congress on Evolutionary Computation, 2003. CEC '03..

[10]  R. Paul Wiegand,et al.  Biasing Coevolutionary Search for Optimal Multiagent Behaviors , 2006, IEEE Transactions on Evolutionary Computation.

[11]  T Haftka Raphael,et al.  Multidisciplinary aerospace design optimization: survey of recent developments , 1996 .

[12]  Nikolaus Hansen,et al.  Evaluating the CMA Evolution Strategy on Multimodal Test Functions , 2004, PPSN.

[13]  Zhen Ji,et al.  DNA Sequence Compression Using Adaptive Particle Swarm Optimization-Based Memetic Algorithm , 2011, IEEE Transactions on Evolutionary Computation.

[14]  P. N. Suganthan,et al.  Differential Evolution Algorithm With Strategy Adaptation for Global Numerical Optimization , 2009, IEEE Transactions on Evolutionary Computation.

[15]  Ahmad Faisal Mohamad Ayob,et al.  A novel evolutionary approach for 2D shape matching based on B-spline modeling , 2011, 2011 IEEE Congress of Evolutionary Computation (CEC).

[16]  Bernhard Sendhoff,et al.  Morphing methods in evolutionary design optimization , 2005, GECCO '05.

[17]  Hans-Paul Schwefel,et al.  Evolution and optimum seeking , 1995, Sixth-generation computer technology series.

[18]  Yuhui Shi,et al.  Biomimicry of parasitic behavior in a coevolutionary particle swarm optimization algorithm for global optimization , 2015, Appl. Soft Comput..

[19]  Bernhard Sendhoff,et al.  Evolutionary Computation Benchmarking Repository , 2006 .

[20]  Amir Hossein Gandomi,et al.  Cuckoo search algorithm: a metaheuristic approach to solve structural optimization problems , 2011, Engineering with Computers.

[21]  X. Yao,et al.  Scaling up fast evolutionary programming with cooperative coevolution , 2001, Proceedings of the 2001 Congress on Evolutionary Computation (IEEE Cat. No.01TH8546).

[22]  Hans-Paul Schwefel,et al.  Evolution and Optimum Seeking: The Sixth Generation , 1993 .

[23]  Ponnuthurai N. Suganthan,et al.  An Adaptive Differential Evolution Algorithm With Novel Mutation and Crossover Strategies for Global Numerical Optimization , 2012, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[24]  Xin Yao,et al.  Drift analysis and average time complexity of evolutionary algorithms , 2001, Artif. Intell..

[25]  Mitchell A. Potter,et al.  The design and analysis of a computational model of cooperative coevolution , 1997 .

[26]  Xin Yao,et al.  Self-adaptive differential evolution with neighborhood search , 2008, 2008 IEEE Congress on Evolutionary Computation (IEEE World Congress on Computational Intelligence).

[27]  Xin Yao,et al.  Evolutionary computation benchmarking repository [Developmental Tools] , 2006, IEEE Computational Intelligence Magazine.

[28]  Bernhard Sendhoff,et al.  Optimisation of a Stator Blade Used in a Transonic Compressor Cascade with Evolution Strategies , 2000 .

[29]  Sean Luke,et al.  Cooperative Multi-Agent Learning: The State of the Art , 2005, Autonomous Agents and Multi-Agent Systems.

[30]  Markus Olhofer,et al.  Advanced high turning compressor airfoils for low Reynolds number condition. Part I: Design and optimization , 2004 .

[31]  Xin Yao,et al.  Large scale evolutionary optimization using cooperative coevolution , 2008, Inf. Sci..

[32]  Weicheng Xie,et al.  A binary differential evolution algorithm learning from explored solutions , 2014, Neurocomputing.

[33]  A Samareh Jamshid,et al.  A Survey of Shape Parameterization Techniques , 1999 .

[34]  Kalyanmoy Deb,et al.  A Computationally Efficient Evolutionary Algorithm for Real-Parameter Optimization , 2002, Evolutionary Computation.

[35]  Bernhard Sendhoff,et al.  A systems approach to evolutionary multiobjective structural optimization and beyond , 2009, IEEE Computational Intelligence Magazine.

[36]  Rainer Storn,et al.  Differential Evolution – A Simple and Efficient Heuristic for global Optimization over Continuous Spaces , 1997, J. Glob. Optim..

[37]  Nikolaus Hansen,et al.  Completely Derandomized Self-Adaptation in Evolution Strategies , 2001, Evolutionary Computation.