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A Class Of Nonlinear Lagrangians: Theory And Algorithm

Author

Listed:
  • LI-WEI ZHANG

    (Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, P. R. China)

  • YONG-HONG REN

    (School of Mathematics, Liaoning Normal University, Dalian 116029, P. R. China)

  • YUE WU

    (School of Management, University of Southampton, Southampton, Highfield Southampton, SO17 1BJ, United Kingdom)

  • XIAN-TAO XIAO

    (Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, P. R. China)

Abstract

This paper establishes a theory framework of a class of nonlinear Lagrangians for solving nonlinear programming problems with inequality constraints. A set of conditions are proposed to guarantee the convergence of nonlinear Lagrangian algorithms, to analyze condition numbers of nonlinear Lagrangian Hessians as well as to develop the dual approaches. These conditions are satisfied by well-known nonlinear Lagrangians appearing in literature. The convergence theorem shows that the dual algorithm based on any nonlinear Lagrangian in the class is locally convergent when the penalty parameter is less than a threshold under a set of suitable conditions on problem functions and the error bound solution, depending on the penalty parameter, is also established. The paper also develops the dual problems based on the proposed nonlinear Lagrangians, and the related duality theorem and saddle point theorem are demonstrated. Furthermore, it is shown that the condition numbers of Lagrangian Hessians at optimal solutions are proportional to the controlling penalty parameters. We report some numerical results obtained by using nonlinear Lagrangians.

Suggested Citation

  • Li-Wei Zhang & Yong-Hong Ren & Yue Wu & Xian-Tao Xiao, 2008. "A Class Of Nonlinear Lagrangians: Theory And Algorithm," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 25(03), pages 327-371.
    • Handle: RePEc:wsi:apjorx:v:25:y:2008:i:03:n:s021759590800178x
    DOI: 10.1142/S021759590800178X
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    Cited by:

    1. Faddoul, R. & Raphael, W. & Chateauneuf, A., 2018. "Maintenance optimization of series systems subject to reliability constraints," Reliability Engineering and System Safety, Elsevier, vol. 180(C), pages 179-188.

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