Numerical Computation of Lightly Multi-Objective Robust Optimal Solutions by Means of Generalized Cell Mapping

In this paper, we present a novel algorithm for the computation of lightly robust optimal solutions for multi-objective optimization problems. To this end, we adapt the generalized cell mapping, originally designed for the global analysis of dynamical systems, to the current context. This is the first time that a set-based method is developed for such kinds of problems. We demonstrate the strength of the novel algorithms on several benchmark problems as well as on one feed-back control design problem where the objectives are given by the peak time, the overshoot, and the absolute tracking error for the linear control system, which has a control time delay. The numerical results indicate that the new algorithm is well-suited for the reliable treatment of low dimensional problems.

[1]  José Francisco Gómez-Aguilar,et al.  A Simple Spectral Observer , 2018 .

[2]  P. J. Zufiria,et al.  Improved Optimal Control Methods Based Upon the Adjoining Cell Mapping Technique , 2003 .

[3]  Gildas Besancon,et al.  Modeling and Simulation of a Hydraulic Network for Leak Diagnosis , 2018, Mathematical and Computational Applications.

[4]  S. Daley,et al.  Optimal-tuning nonlinear PID control of hydraulic systems , 2000 .

[5]  Bogoya,et al.  The Averaged Hausdorff Distances in Multi-Objective Optimization: A Review , 2019, Mathematics.

[6]  Jian-Qiao Sun,et al.  Non-Epsilon Dominated Evolutionary Algorithm for the Set of Approximate Solutions , 2020, Mathematical and Computational Applications.

[7]  Matthias Ehrgott,et al.  Minmax robustness for multi-objective optimization problems , 2014, Eur. J. Oper. Res..

[8]  M. Dellnitz,et al.  A subdivision algorithm for the computation of unstable manifolds and global attractors , 1997 .

[9]  El-Ghazali Talbi,et al.  Archivers for the representation of the set of approximate solutions for MOPs , 2018, J. Heuristics.

[10]  Yousef Naranjani,et al.  Simple cell mapping method for multi-objective optimal feedback control design , 2013, International Journal of Dynamics and Control.

[11]  Carlos Hernández,et al.  Parallel simple cell mapping for multi-objective optimization , 2016 .

[12]  Anita Schöbel,et al.  Robustness for uncertain multi-objective optimization: a survey and analysis of different concepts , 2016, OR Spectr..

[13]  P. Loridan.  ε-solutions in vector minimization problems , 1984 .

[14]  Oliver Schütze,et al.  A (p,q)-Averaged Hausdorff Distance for Arbitrary Measurable Sets , 2018, Mathematical and Computational Applications.

[15]  Günter Rudolph,et al.  Capabilities of EMOA to Detect and Preserve Equivalent Pareto Subsets , 2007, EMO.

[16]  Zhi-Chang Qin,et al.  Multi-objective optimal design of sliding mode control with parallel simple cell mapping method , 2017 .

[17]  C. S. Hsu,et al.  Cell-to-Cell Mapping , 1987 .

[18]  Gábor Csernák,et al.  Clustered Simple Cell Mapping: An extension to the Simple Cell Mapping method , 2017, Commun. Nonlinear Sci. Numer. Simul..

[19]  Zhi-Chang Qin,et al.  Multi-objective optimal design of feedback controls for dynamical systems with hybrid simple cell mapping algorithm , 2014, Commun. Nonlinear Sci. Numer. Simul..

[20]  M. Dellnitz,et al.  Covering Pareto Sets by Multilevel Subdivision Techniques , 2005 .

[21]  Günter Rudolph,et al.  Pareto Set and EMOA Behavior for Simple Multimodal Multiobjective Functions , 2006, PPSN.

[22]  Mohammed Mazen Alhato,et al.  Direct Power Control Optimization for Doubly Fed Induction Generator Based Wind Turbine Systems , 2019, Mathematical and Computational Applications.

[23]  Michael Dellnitz,et al.  A Survey of Recent Trends in Multiobjective Optimal Control—Surrogate Models, Feedback Control and Objective Reduction , 2018, Mathematical and Computational Applications.

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

[25]  Jörg Fliege,et al.  Robust multiobjective optimization & applications in portfolio optimization , 2014, Eur. J. Oper. Res..

[26]  Carlos A. Coello Coello,et al.  Using the Averaged Hausdorff Distance as a Performance Measure in Evolutionary Multiobjective Optimization , 2012, IEEE Transactions on Evolutionary Computation.

[27]  Bernhard Sendhoff,et al.  Robust Optimization - A Comprehensive Survey , 2007 .

[28]  Alicia Cordero,et al.  Modified Potra-Pták Multi-step Schemes with Accelerated Order of Convergence for Solving Systems of Nonlinear Equations , 2018, Mathematical and Computational Applications.

[29]  Carlos Hernández,et al.  Global Multi-objective Optimization by Means of Cell Mapping Techniques , 2017 .

[30]  Fu-Rui Xiong,et al.  Finding zeros of nonlinear functions using the hybrid parallel cell mapping method , 2016, Commun. Nonlinear Sci. Numer. Simul..