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DC Programming Approaches for BMI and QMI Feasibility Problems

  • Conference paper
Advanced Computational Methods for Knowledge Engineering

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 282))

Abstract

We propose some new DC (difference of convex functions) programming approaches for solving the Bilinear Matrix Inequality (BMI) Feasibility Problems and the Quadratic Matrix Inequality (QMI) Feasibility Problems. They are both important NP-hard problems in the field of robust control and system theory. The inherent difficulty lies in the nonconvex set of feasible solutions. In this paper, we will firstly reformulate these problems as a DC program (minimization of a concave function over a convex set). Then efficient approaches based on the DC Algorithm (DCA) are proposed for the numerical solution. A semidefinite program (SDP) is required to be solved during each iteration of our algorithm. Moreover, a hybrid method combining DCA with an adaptive Branch and Bound is established for guaranteeing the feasibility of the BMI and QMI. A concept of partial solution of SDP via DCA is proposed to improve the convergence of our algorithm when handling more large-scale cases. Numerical simulations of the proposed approaches and comparison with PENBMI are also reported.

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Author information

Authors and Affiliations

  1. University of Paris 6, Paris, France

    Yi-Shuai Niu

  2. National Institute for Applied Sciences, Rouen, France

    Tao Pham Dinh

Authors
  1. Yi-Shuai Niu
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  2. Tao Pham Dinh
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Correspondence to Yi-Shuai Niu .

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Editors and Affiliations

  1. Department of Networked Systems and Services, Budapest University of Technology and Economics, Budapest, Hungary

    Tien van Do

  2. Ile du Saulcy, LITA, UFR MIM, Metz, France

    Hoai An Le Thi

  3. Institute of Informatics, Wrocław University of Technology, Wrocław, Poland

    Ngoc Thanh Nguyen

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Niu, YS., Dinh, T.P. (2014). DC Programming Approaches for BMI and QMI Feasibility Problems. In: van Do, T., Thi, H., Nguyen, N. (eds) Advanced Computational Methods for Knowledge Engineering. Advances in Intelligent Systems and Computing, vol 282. Springer, Cham. https://doi.org/10.1007/978-3-319-06569-4_3

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  • DOI: https://doi.org/10.1007/978-3-319-06569-4_3

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-06568-7

  • Online ISBN: 978-3-319-06569-4

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