DC Programming Approaches for BMI and QMI Feasibility Problems

  • Yi-Shuai NiuEmail author
  • Tao Pham Dinh
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 282)


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.


BMI/QMI DC program DCA Branch and Bound SDP 


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  1. 1.
    Benson, S.T., Ye, Y.Y.: DSDP: A complete description of the algorithm and a proof of convergence can be found in Solving Large-Scale Sparse Semidefinite Programs for Combinatorial Optimization. SIAM Journal on Optimization 10(2), 443–461 (2000), CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Beran, E.B., Vandenberghe, L., Boyd, S.: A global BMI algorithm based on the generalized Benders decomposition. In: Proceedings of the European Control Conference, Brussels, Belgium (July 1997)Google Scholar
  3. 3.
    Borchers, B.: CSDP: a C library for semidefinite programming, Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM (November 1998),
  4. 4.
    Floudas, C.A., Visweswaran, V.: A primal-relaxed dual global optimization approach. Journal of Optimization Theory and Applications 78, 187–225 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Fujioka, H., Hoshijima, K.: Bounds for the BMI eingenvalue problem - a good lower bound and a cheap upper bound. Transactions of the Society of Instrument and Control Engineers 33, 616–621 (1997)Google Scholar
  6. 6.
    Fujisawa, K., Kojima, M., Nakata, K.: SDPA (SemiDefinite Programming Algorithm) - user’s manual - version 6.20. Research Report B-359, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo, Japan (January 2005), (revised May 2005)
  7. 7.
    Fukuda, M., Kojima, M.: Branch-and-Cut Algorithms for the Bilinear Matrix Inequality Eigenvalue Problem. Computational Optimization and Applications 19(1), 79–105 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Goh, K.C., Safonov, M.G., Papavassilopoulos, G.P.: Global optimization for the biaffine matrix inequality problem. Journal of Global Optimization 7, 365–380 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Goh, K.C., Safonov, M.G., Ly, J.H.: Robust synthesis via bilinear matrix inequalities. International Journal of Robust and Nonlinear Control 6, 1079–1095 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Horst, R.: D.C. Optimization: Theory, Methods and Algorithms. In: Horst, R., Pardalos, P.M. (eds.) Handbook of Global Optimization, pp. 149–216. Kluwer Academic Publishers, Dordrecht (1995)CrossRefGoogle Scholar
  11. 11.
    Horst, R., Thoai, N.V.: DC Programming: Overview. Journal of Optimization Theory and Applications 103, 1–43 (1999)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Hiriart Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms. Springer, Heidelberg (1993)Google Scholar
  13. 13.
    Horst, R., Pardalos, P.M., Thoai, N.V.: Introduction to Global Optimization, 2nd edn. Kluwer Academic Publishers, Netherlands (2000)CrossRefzbMATHGoogle Scholar
  14. 14.
    Kawanishi, M., Sugie, T., Kanki, H.: BMI global optimization based on branch and bound method taking account of the property of local minima. In: Proceedings of the Conference on Decision and Control, San Diego, CA (December 1997)Google Scholar
  15. 15.
    Kočvara, M., Stingl, M.: PENBMI User’s Guide (Version 2.1) (February 16, 2006)Google Scholar
  16. 16.
    Le Thi, H.A., Pham Dinh, T.: Solving a class of linearly constrained indefinite quadratic problems by DC Algorithms. Journal of Global Optimization 11, 253–285 (1997)CrossRefzbMATHGoogle Scholar
  17. 17.
    Le Thi, H.A., Pham Dinh, T., Le Dung, M.: Exact penalty in d.c. programming. Vietnam Journal of Mathematics 27(2), 169–178 (1999)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Le Thi, H.A.: An efficient algorithm for globally minimizing a quadratic function under convex quadratic constraints. Mathematical Programming Ser. A. 87(3), 401–426 (2000)CrossRefzbMATHGoogle Scholar
  19. 19.
    Le Thi, H.A., Pham Dinh, T.: A continuous approach for large-scale constrained quadratic zero-one programming (In honor of Professor ELSTER, Founder of the Journal Optimization). Optimization 45(3), 1–28 (2001)Google Scholar
  20. 20.
    Le Thi, H.A., Pham Dinh, T.: Large Scale Molecular Optimization From Distance Matrices by a D.C. Optimization Approach. SIAM Journal on Optimization 4(1), 77–116 (2003)Google Scholar
  21. 21.
    Le Thi, H.A.: Solving large scale molecular distance geometry problems by a smoothing technique via the gaussian transform and d.c. programming. Journal of Global Optimization 27(4), 375–397 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Le Thi, H.A., Pham Dinh, T., François, A.: Combining DCA and Interior Point Techniques for large-scale Nonconvex Quadratic Programming. Optimization Methods & Software 23(4), 609–629 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Liu, S.M., Papavassilopoulos, G.P.: Numerical experience with parallel algorithms for solving the BMI problem. In: 13th Triennial World Congress of IFAC, San Francisco, CA (July 1996)Google Scholar
  24. 24.
    Löfberg, J.: YALMIP: A Toolbox for Modeling and Optimization in MATLAB. In: Proceedings of the CACSD Conference, Taipei, Taiwan (2004),
  25. 25.
    MATLAB R2007a: Documentation and User Guides,
  26. 26.
    Mesbahi, M., Papavassilopoulos, G.P.: A cone programming approach to the bilinear matrix inequality problem and its geometry. Mathematical Programming 77, 247–272 (1997)zbMATHMathSciNetGoogle Scholar
  27. 27.
    Mittelmann, H.D.: Several SDP-codes on problems from SDPLIB,
  28. 28.
    Niu, Y.S., Pham Dinh, T.: A DC Programming Approach for Mixed-Integer Linear Programs. In: Le Thi, H.A., Bouvry, P., Pham Dinh, T. (eds.) MCO 2008. CCIS, vol. 14, pp. 244–253. Springer, Heidelberg (2008)Google Scholar
  29. 29.
    Niu, Y.S.: DC programming and DCA combinatorial optimization and polynomial optimization via SDP techniques, National Institute of Applied Sciences, Rouen, France (2010)Google Scholar
  30. 30.
    Niu, Y.S., Pham Dinh, T.: An Efficient DC Programming Approach for Portfolio Decision with Higher Moments. Computational Optimization and Applications 50(3), 525–554 (2010)MathSciNetGoogle Scholar
  31. 31.
    Niu, Y.S., Pham Dinh, T.: Efficient DC programming approaches for mixed-integer quadratic convex programs. In: Proceedings of the International Conference on Industrial Engineering and Systems Management (IESM 2011), Metz, France, pp. 222–231 (2011)Google Scholar
  32. 32.
    Niu, Y.S., Pham Dinh, T., Le Thi, H.A., Judice, J.J.: Efficient DC Programming Approaches for the Asymmetric Eigenvalue Complementarity Problem. Optimization Methods and Software 28(4), 812–829 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Pham Dinh, T., Le Thi, H.A.: Convex analysis approach to D.C. programming: Theory, Algorithms and Applications. Acta Mathematica Vietnamica 22(1), 289–355 (1997)zbMATHMathSciNetGoogle Scholar
  34. 34.
    Pham Dinh, T., Le Thi, H.A.: DC optimization algorithms for solving the trust region subproblem. SIAM J. Optimization 8, 476–507 (1998)CrossRefzbMATHGoogle Scholar
  35. 35.
    Pham Dinh, T., Le Thi, H.A.: DC Programming. Theory, Algorithms, Applications: The State of the Art. In: First International Workshop on Global Constrained Optimization and Constraint Satisfaction, Nice, October 2-4 (2002)Google Scholar
  36. 36.
    Pham Dinh, T., Le Thi, H.A.: The DC programming and DCA Revisited with DC Models of Real World Nonconvex Optimization Problems. Annals of Operations Research 133, 23–46 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, N.J. (1970)zbMATHGoogle Scholar
  38. 38.
    Safonov, M.G., Goh, K.C., Ly, J.H.: Control system synthesis via bilinear matrix inequalities. In: Proceedings of the American Control Conference, Baltimore, MD (June 1994)Google Scholar
  39. 39.
    Sherali, H.D., Alameddine, A.R.: A new reformulation-linearization technique for bilinear programming problems. Journal of Global Optimization 2, 379–410 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  40. 40.
    Sturm, J.F.: SeDuMi 1.2: a MATLAB toolbox for optimization over symmetric cones. Department of Quantitative Economics, Maastricht University, Maastricht, The Netherlands (August 1998),
  41. 41.
    Takano, S., Watanabe, T., Yasuda, K.: Branch and bound technique for global solution of BMI. Transactions of the Society of Instrument and Control Engineers 33, 701–708 (1997)Google Scholar
  42. 42.
    Toker, O., Özbay, H.: On the NP-hardness of solving bilinear matrix inequalities and simultaneous stabilization with static output feedback. In: American Control Conference, Seattle, WA (1995)Google Scholar
  43. 43.
    Tuan, H.D., Hosoe, S., Tuy, H.: D.C. optimization approach to robust controls: Feasibility problems. IEEE Transactions on Automatic Control 45, 1903–1909 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  44. 44.
    Tuan, H.D., Apkarian, P., Nakashima, Y.: A new Lagrangian dual global optimization algorithm for solving bilinear matrix inequalities. International Journal of Robust and Nonlinear Control 10, 561–578 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  45. 45.
    Van Antwerp, J.G.: Globally optimal robust control for systems with time-varying nonlinear perturbations. Master thesis, University of Illinois at Urbana-Champaign, Urbana, IL (1997)Google Scholar
  46. 46.
    Wolkowicz, H., Saigal, R., Vandenberghe, L.: Handbook of Semidefinite Programming - Theory, Algorithms, and Applications. Kluwer Academic Publishers, USA (2000)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.University of Paris 6ParisFrance
  2. 2.National Institute for Applied SciencesRouenFrance

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