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Multi-objective convex polynomial optimization and semidefinite programming relaxations

Author

Listed:
  • Jae Hyoung Lee

    (Pukyong National University)

  • Nithirat Sisarat

    (Naresuan University)

  • Liguo Jiao

    (Soochow University)

Abstract

This paper aims to find efficient solutions to a multi-objective optimization problem (MP) with convex polynomial data. To this end, a hybrid method, which allows us to transform problem (MP) into a scalar convex polynomial optimization problem $$({\mathrm{P}}_{z})$$ ( P z ) and does not destroy the properties of convexity, is considered. First, we show an existence result for efficient solutions to problem (MP) under some mild assumption. Then, for problem $$(P_{z})$$ ( P z ) , we establish two kinds of representations of non-negativity of convex polynomials over convex semi-algebraic sets, and propose two kinds of finite convergence results of the Lasserre-type hierarchy of semidefinite programming relaxations for problem $$({\mathrm{P}}_{z})$$ ( P z ) under suitable assumptions. Finally, we show that finding efficient solutions to problem (MP) can be achieved successfully by solving hierarchies of semidefinite programming relaxations and checking a flat truncation condition.

Suggested Citation

  • Jae Hyoung Lee & Nithirat Sisarat & Liguo Jiao, 2021. "Multi-objective convex polynomial optimization and semidefinite programming relaxations," Journal of Global Optimization, Springer, vol. 80(1), pages 117-138, May.
    • Handle: RePEc:spr:jglopt:v:80:y:2021:i:1:d:10.1007_s10898-020-00969-x
    DOI: 10.1007/s10898-020-00969-x
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    References listed on IDEAS

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    1. Victor Blanco & Justo Puerto & Safae El Haj Ben Ali, 2014. "A Semidefinite Programming approach for solving Multiobjective Linear Programming," Journal of Global Optimization, Springer, vol. 58(3), pages 465-480, March.
    2. Gorissen, B.L. & den Hertog, D., 2012. "Approximating the Pareto Set of Multiobjective Linear Programs via Robust Optimization," Other publications TiSEM 666c5307-4a4e-4be4-a0d0-b, Tilburg University, School of Economics and Management.
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    4. Jae Hyoung Lee & Liguo Jiao, 2018. "Solving Fractional Multicriteria Optimization Problems with Sum of Squares Convex Polynomial Data," Journal of Optimization Theory and Applications, Springer, vol. 176(2), pages 428-455, February.
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    6. Gorissen, B.L. & den Hertog, D., 2012. "Approximating the Pareto Set of Multiobjective Linear Programs via Robust Optimization," Discussion Paper 2012-031, Tilburg University, Center for Economic Research.
    7. Dinh The Luc, 2016. "Multiobjective Linear Programming," Springer Books, Springer, edition 1, number 978-3-319-21091-9, December.
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    Cited by:

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    2. Feng Guo & Liguo Jiao, 2023. "A new scheme for approximating the weakly efficient solution set of vector rational optimization problems," Journal of Global Optimization, Springer, vol. 86(4), pages 905-930, August.

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