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On Polynomial Optimization Over Non-compact Semi-algebraic Sets

Author

Listed:
  • V. Jeyakumar

    (University of New South Wales)

  • J. B. Lasserre

    (LAAS-CNRS and Institute of Mathematics)

  • G. Li

    (University of New South Wales)

Abstract

The optimal value of a polynomial optimization over a compact semi-algebraic set can be approximated as closely as desired by solving a hierarchy of semidefinite programs and the convergence is finite generically under the mild assumption that a quadratic module generated by the constraints is Archimedean. We consider a class of polynomial optimization problems with non-compact semi-algebraic feasible sets, for which an associated quadratic module, that is generated in terms of both the objective function and the constraints, is Archimedean. For such problems, we show that the corresponding hierarchy converges and the convergence is finite generically. Moreover, we prove that the Archimedean condition (as well as a sufficient coercivity condition) can be checked numerically by solving a similar hierarchy of semidefinite programs. In other words, under reasonable assumptions, the now standard hierarchy of semidefinite programming relaxations extends to the non-compact case via a suitable modification.

Suggested Citation

  • V. Jeyakumar & J. B. Lasserre & G. Li, 2014. "On Polynomial Optimization Over Non-compact Semi-algebraic Sets," Journal of Optimization Theory and Applications, Springer, vol. 163(3), pages 707-718, December.
    • Handle: RePEc:spr:joptap:v:163:y:2014:i:3:d:10.1007_s10957-014-0545-3
    DOI: 10.1007/s10957-014-0545-3
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    References listed on IDEAS

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    1. Laurent, M., 2009. "Sums of squares, moment matrices and optimization over polynomials," Other publications TiSEM 9fef820b-69d2-43f2-a501-e, Tilburg University, School of Economics and Management.
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    Cited by:

    1. Vaithilingam Jeyakumar & Guoyin Li, 2017. "Exact Conic Programming Relaxations for a Class of Convex Polynomial Cone Programs," Journal of Optimization Theory and Applications, Springer, vol. 172(1), pages 156-178, January.
    2. T. D. Chuong & V. Jeyakumar, 2018. "Generalized Lagrangian duality for nonconvex polynomial programs with polynomial multipliers," Journal of Global Optimization, Springer, vol. 72(4), pages 655-678, December.
    3. T. D. Chuong & V. Jeyakumar, 2017. "Finding Robust Global Optimal Values of Bilevel Polynomial Programs with Uncertain Linear Constraints," Journal of Optimization Theory and Applications, Springer, vol. 173(2), pages 683-703, May.
    4. Trang T. Du & Toan M. Ho, 2019. "Polynomial Optimization on Some Unbounded Closed Semi-algebraic Sets," Journal of Optimization Theory and Applications, Springer, vol. 183(1), pages 352-363, October.
    5. Sönke Behrends & Anita Schöbel, 2020. "Generating Valid Linear Inequalities for Nonlinear Programs via Sums of Squares," Journal of Optimization Theory and Applications, Springer, vol. 186(3), pages 911-935, September.

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