Interactive bundle-based method for nondifferentiable multiobjective optimization: NIMBUS.

*(English)*Zbl 0855.90114Summary: NIMBUS, an interactive method for nondifferentiable multiobjective optimization problems, is described. The algorithm is based on the classification of objective functions. At each iteration, a decision maker is asked to classify the objective functions into up to five different classes: those to be improved, those to be improved till some aspiration level, those to be accepted as they are, those to be impaired till some bound, and those allowed to change freely.

According to the classification, a new (multiobjective) optimization problem is formed, which is solved by an MPB (Multiobjective Proximal Bundle) method. The MPB method is a generalization of Kiwiel’s proximal bundle approach for nondifferentiable single objective optimization into the multiobjective case. The multiple objective functions are treated individually without employing any scalarization. The method is capable of handling several nonconvex locally Lipschitz continuous objective functions subject to nonlinear (possibly nondifferentiable) constraints.

Finally, numerical experiments with two academic test problems are reported. The aim is to briefly demonstrate how the method works.

According to the classification, a new (multiobjective) optimization problem is formed, which is solved by an MPB (Multiobjective Proximal Bundle) method. The MPB method is a generalization of Kiwiel’s proximal bundle approach for nondifferentiable single objective optimization into the multiobjective case. The multiple objective functions are treated individually without employing any scalarization. The method is capable of handling several nonconvex locally Lipschitz continuous objective functions subject to nonlinear (possibly nondifferentiable) constraints.

Finally, numerical experiments with two academic test problems are reported. The aim is to briefly demonstrate how the method works.

##### Keywords:

multiobjective proximal bundle method; NIMBUS; nondifferentiable multiobjective optimization
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\textit{K. Miettinen} and \textit{M. M. Mäkelä}, Optimization 34, No. 3, 231--246 (1995; Zbl 0855.90114)

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##### References:

[1] | Benayon R., Linear Programming with Multiple Objective Functions: Step Method (STEM) 1 (3) pp 366– (1971) |

[2] | Clarke F.H., Optirnization and Vonsmooth Analjsis (1983) |

[3] | Hwang C.L., Multiple Objectice Decision Making-Methods and Applications: A State-of-the-Art Survey (1979) · doi:10.1007/978-3-642-45511-7 |

[4] | Kiwiel, K.C. 1984.An Aggregate Subgradient Descent Method for Solving Large Concex Non-smooth Multiobjective Minimization Problems, Edited by: Straszak, A. 283–288. Pergamon Press. Large Scale Systems: Theory and Applications 1983 |

[5] | Kiwiel K.C., A Descent Method for Nonsmooth concex Multiobjective Minimization 8 (2) pp 119– (1985) · Zbl 0564.90068 |

[6] | Kiwiel K.C., Methods of Descent for Nondifferentiable Optirnization (1985) · Zbl 0561.90059 |

[7] | Kiwiel K.C., A Method for Solcing Certain Quadratic Programming Problems Arising in Non-smoorh Optimization 6 pp 137– (1986) |

[8] | Kiwiel K.C., Proximity Control in Bundle Methods for concex Nondifferentiable Optimization 46 pp 105– (1990) · Zbl 0697.90060 |

[9] | Mäkelä M.M., Issues oflmplementing a Fortran Subroutine Package NSOLIB for Nonsmooth Optimization (1993) |

[10] | Mäkelä M.M., Nonsmooth Optirnization: Analysis and Algorithms with Applications to Optimal Contro (1992) · doi:10.1142/1493 |

[11] | Miettinen K., On the Methodology of Multiobjective Optimization with Applications (1994) · Zbl 0831.90099 |

[12] | Miettinen K., An Interactive Method for Nonsmooth Optirnization with Application to optimal Control 2 pp 31– (1993) |

[13] | Miettinen, K. and Mäkelä, M.M. 1994. A Nondifferenriable Multiple Criteria Optimization Method Applied to Continuous Casting Process. Proceedings of the Seventh European Conference on Mathematics in Industry. 1994. Edited by: Fasano, A. and Primicerio, M.B.G. pp.255–262. Teubner Stuttgart. |

[14] | Mukai H., Algorithms for Multicriterion Optimization 25 (2) pp 177– (1980) · Zbl 0428.90072 |

[15] | Nakayama H., Satisjcing Trade-Off ,Method for Multiobjective Programming pp 113– (1984) |

[16] | Narula S.C., A Flexible Method for Nonlinear Multicriteria Decisionmaking Problems 19 (4) pp 883– (1989) |

[17] | Rockafellar R.T., Lagrange Multipliers and Subdericatives of Optimal Value Functions in Nonlinear Programming 17 (4) pp 28– (1982) · Zbl 0478.90060 |

[18] | Wang S., Algorithms for Multiobjective and Nonsmooth Optimization pp 131– (1989) |

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