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I'm looking for papers dealing with multi-objective non-linear programming which could help me implement an algorithm to solve my problem.

My problem is :

Maximize $f(x) = c \cdot x$, while minimizing $g(x) = r \cdot x$, where $\cdot$ is the scalar product.

Constrained by $h_1(x) = \frac{v \cdot x}{p \cdot x} \geq 0.5$ and $h_2(x) = b \cdot x = B$

Given that $b, c, p, r, v \in (\mathbb{N}^*)^n, B \in \mathbb{N}^*, x \in [0, 1]^n$

For me, $n$ will be in the order of $1000$.

I know there is no unique solution so I'm looking for the set of Pareto optimal solutions, therefore I am not interested in scalarizing. I could use interactive methods though.

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In Gandibleux, X. (Ed.). (2006). Multiple criteria optimization: state of the art annotated bibliographic surveys (Vol. 52). Springer Science & Business Media., I found in chapter 5 many interactive nonlinear multiobjective procedures explained. According to Kaisa Miettinen, none is better than another and they all have their pros and cons.

We can cite :

  • Interactive Surrogate Worth Trade-Off Method [1]
  • Geoffrion-Dyer-Feinberg Method [2]
  • Tchebycheff Method [3]
  • Step Method [4]
  • Reference Point Method [5]
  • GUESS Method [6]
  • Satisficing Trade-Off Method [7]
  • Light Beam Search [8]
  • Reference Direction Approach [9]
  • Reference Direction Method [10]
  • NIMBUS Method [11]

    [1]: V. Chankong and Y. Y. Haimes. Multiobjective Decision Making Theory and Methodology. Elsevier Science Publishing Co., New York, 1983

    [2]: A. M. Geoffrion, J. S. Dyer, and A. Feinberg. An interactive approach for multi-criterion optimization, with an application to the operation of an academic department. Management Science, 19:357–368, 1972

    [3]: R. E. Steuer. The Tchebycheff procedure of interactive multiple objective programming. In B. Karpak and S. Zionts, editors, Multiple Criteria Decision Making and Risk Analysis Using Microcomputers, pages 235–249. Springer-Verlag, Berlin, Heidelberg, 1989.

    [4]: R. Benayoun, J. de Montgolfier, J. Tergny, and O. Laritchev. Linear programming with multiple objective functions: Step method (STEM). Mathematical Programming, 1:366–375, 1971

    [5]: A. P. Wierzbicki. The use of reference objectives in multiobjective optimization. In G. Fandel and T. Gal, editors, Multiple Criteria Decision Making Theory and Applications, pages 468–486. Springer-Verlag, Berlin, Heidelberg, 1980

    [6]: J. T. Buchanan. A naïve approach for solving MCDM problems: The GUESS method. Journal of the Operational Research Society, 48:202–206, 1997.

    [7]: H. Nakayama. Aspiration level approach to interactive multi- objective programming and its applications. In P. M. Pardalos, Y. Siskos, and C. Zopounidis, editors, Advances in Multicriteria Analysis, pages 147–174. Kluwer Academic Publishers, Dordrecht, 1995

    [8]: A. Jaszkiewicz The light beam search – outranking based interactive procedure for multiple-objective mathematical programming. In P. M. Pardalos, Y. Siskos, and C. Zopounidis, editors, Advances in Multicriteria Analysis, pages 129–146. Kluwer Academic Publishers, Dordrecht, 1995.

    [9]: P. Korhonen. Reference direction approach to multiple objective linear programming: Historical overview. In M. H. Karwan, J. Spronk, and J. Wallenius, editors, Essays in Decision Making: A Volume in Honour of Stanley Zionts, pages 74–92. Springer- Verlag, Berlin, Heidelberg, 1997

    [10]: S. C. Narula, L. Kirilov, and V. Vassilev. An interactive algorithm for solving multiple objective nonlinear programming problems. In G. H. Tzeng, H. F. Wand, U. P. Wen, and P. L. Yu, editors, Multiple Criteria Decision Making – Proceedings of the Tenth International Conference: Expand and Enrich the Domains of Thinking and Application, pages 119–127. Springer-Verlag, New York, 1994

    [11]: K. Miettinen. Nonlinear Multiobjective Optimization. Kluwer Academic Publishers, Boston, 1999

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