Abstract
The optimization problem with the Bilinear Matrix Inequality (BMI) is one of the problems which have greatly interested researchers of system and control theory in the last few years. This inequality permits to reduce in an elegant way various problems of robust control into its form. However, in contrast to the Linear Matrix Inequality (LMI), which can be solved by interior-point-methods, the BMI is a computationally difficult object in theory and in practice. This article improves the branch-and-bound algorithm of Goh, Safonov and Papavassilopoulos (Journal of Global Optimization, vol. 7, pp. 365–380, 1995) by applying a better convex relaxation of the BMI Eigenvalue Problem (BMIEP), and proposes new Branch-and-Bound and Branch-and-Cut Algorithms. Numerical experiments were conducted in a systematic way over randomly generated problems, and they show the robustness and the efficiency of the proposed algorithms.
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References
F.A. Al-Khayyal and J.E. Falk, “Jointly constrained biconvex programming, ” Mathematics of Operational Research, vol. 8, pp. 273–286, 1983.
E.B. Beran, “Methods for optimization-based fixed-order control design, ” PhD thesis, Department of Mathematics and Department of Automation, Technical University of Denmark, Denmark, September 1997.
E.B. Beran, L. Vandenberghe, and S. Boyd, “A global BMI algorithm based on the generalized Benders decomposition, ” in Proceedings of the European Control Conference, Brussels, Belgium, July 1997.
B. Borchers, “CSDP, 2.3 user's guide, ” Optimization Methods and Software, vol. 11–12, pp. 597–611, 1999. Available at http://www.nmt.edu/~borchers/csdp.html.
S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM: Philadelphia, 1994.
M.M. Deza and M. Laurent, Geometry of Cuts and Metrics, Springer-Verlag: Berlin, 1997.
L.El Ghaoui and V. Balakrishnan, “Synthesis of fixed-structure controllers via numerical optimization, ” in Proceedings of the Conference on Decision and Control, Lake Buena Vista, FL, December 1994.
C.A. Floudas and V. Visweswaran, “A primal-relaxed dual global optimization approach, ” Journal of Optimization Theory and Applications, vol. 78, pp. 187–225, 1993.
H. Fujioka and K. Hoshijima, “Bounds for the BMI eingenvalue problem—a good lower bound and a cheap upper bound, ” Transactions of the Society of Instrument and Control Engineers, vol. 33, pp. 616–621, 1997.
K. Fujisawa, M. Kojima, and K. Nakata, “SDPA (SemiDefinite Programming Algorithm)—user's manual—, ” Research Report B-308, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo, Japan, December 1995, revised May 1998. Available at ftp://ftp.is.titech.ac.jp/ pub/OpRes/software/SDPA.
K. Fukuda, “Cdd/Cdd+ reference manual, ” Institute for Operations Research, ETH-Zentrum, Zurich, Switzerland, December 1997. Available at http://www.ifor.math.ethz.ch/ifor/staff/fukuda/cdd home/cdd.html.
M. Fukuda, “Branch-and-cut algorithms for bilinear matrix inequality problems, ” Master thesis, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo, Japan, February 1999.
K.-C. Goh, “Robust control synthesis via bilinear matrix inequalities, ” PhD thesis, University of Southern California, Los Angeles, CA, May 1995.
K.-C. Goh, M.G. Safonov, and J.H. Ly, “Robust synthesis via bilinear matrix inequalities, ” International Journal of Robust and Nonlinear Control, vol. 6, pp. 1079–1095, 1996.
K.-C. Goh, M.G. Safonov, and G.P. Papavassilopoulos, “Global optimization for the biaffine matrix inequality problem, ” Journal of Global Optimization, vol. 7, pp. 365–380, 1995.
P. Huard, “Resolution of mathematical programming with nonlinear constraints by the method of centres, ” in Nonlinear Programming, J. Abadie(ed.), North-Holland Publishing Company: Amsterdam, 1967.
F. Jarre, “A QQP-minimization method for semidefinite and smooth nonconvex programs, ” Working Paper, Abteilung Mathematik, Universit¨at Trier, Trier, Germany, August 1998, revised December 1999.
M. Kawanishi, T. Sugie, and H. Kanki, “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.
M. Kojima and L. Tun¸cel, “Discretization and localization in successive convex relaxation method for nonconvex quadratic optimization, ” Mathematical Programming, vol. 89, pp. 79–111, 2000.
M. Kojima and L. Tun¸cel, “Cones of matrices and successive convex relaxations of nonconvex sets, ” SIAM Journal on Optimization, vol. 10, pp. 750–778, 2000.
S.-M. Liu and G.P. Papavassilopoulos, “Numerical experience with parallel algorithms for solving the BMI problem, ” in 13th Triennial World Congress of IFAC, San Francisco, CA, July 1996.
M. Mesbahi and G.P. Papavassilopoulos, “A cone programming approach to the bilinear matrix inequality problem and its geometry, ” Mathematical Programming, vol. 77, pp. 247–272, 1997.
Y. Nesterov and A. Nemirovskii, Interior-Point Polynomial Algorithms in Convex Programming, SIAM: Philadelphia, 1994.
K.G. Ramakrishnan, M.G.C. Resende, and P.M. Pardalos, “A branch and bound algorithm for the quadraticr assignment problem using a lower bound based on linear programming, ” in State of the Art in Global Optimization, C.A. Floudas and P.M. Pardalos (eds.), Kluwer Academic Publishers: Dordrecht, 1996.
M.G. Safonov, K.-C. Goh, and J.H. Ly, “Control system synthesis via bilinear matrix inequalities, ” in Proceedings of the American Control Conference, Baltimore, MD, June 1994.
H.D. Sherali and A. Alameddine, “A new reformulation-linearization technique for bilinear programming problems, ” Journal of Global Optimization, vol. 2, pp. 379–410, 1992.
J.F. Sturm, “Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones, ” Optimization Methods and Software, vol. 11–12, pp. 625–653, 1999. Available at http://fewcal.kub.nl/sturm/software/ sedumi.html.
S. Takano, T. Watanabe, and K. Yasuda, “Branch and bound technique for global solution of BMI, ”Transactions of the Society of Instrument and Control Engineers, vol. 33, pp. 701–708, 1997 (in Japanese).
K.C. Toh, M.J. Todd, and R.H. T¨ut¨unc¨u, “SDPT3—a MATLAB software package for semidefinite programming, Version 1.3, ” Optimization Methods and Software, vol. 11–12, pp. 545–581, 1999. Available at http://www.math.nus.edu.sg/~mattohkc/.
O. Toker and H. Ozbay, “On the NP-hardness of solving bilinear matrix inequalities and simultaneous stabilization with static output feedback, ” in Proceedings of the American Control Conference, Seattle, WA, June 1995.
H.D. Tuan, S. Hosoe, and H. Tuy, “D.C. optimization approach to robust control: Feasibility problems, ” IEEE Transactions on Automatic Control, vol. 45, pp. 1903–1909, 2000.
J.G. VanAntwerp, R.D. Braatz, and N.V. Sahinidis, “Globally optimal robust control for systems with timevarying nonlinear perturbations, ” Computers & Chemical Engineering, vol. 21, pp. S125–S130, 1997.
L. Vandenberghe and S. Boyd, “Semidefinite Programming, ” SIAM Review, vol. 38, pp. 49–95, 1996.
Y. Wakasa, M. Sasaki, and T. Tanino, “A primal-relaxed dual global optimization approach for the BMI problem, ” in Proceedings of the 26th Symposium of Control Theory, Chiba, Japan, May 1997 (in Japanese).
Y. Yajima, M.V. Ramana, and P.M. Pardalos, “Cuts and semidefinite relaxations for nonconvex quadratic problems, ” Research Report 98-1, Department of Industrial Engineering and Management, Tokyo Institute of Technology, Tokyo, Japan, January 1998.
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Fukuda, M., Kojima, M. Branch-and-Cut Algorithms for the Bilinear Matrix Inequality Eigenvalue Problem. Computational Optimization and Applications 19, 79–105 (2001). https://doi.org/10.1023/A:1011224403708
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DOI: https://doi.org/10.1023/A:1011224403708