Global Optimization pp 155-210

Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 84)

Disciplined Convex Programming

  • Michael Grant
  • Stephen Boyd
  • Yinyu Ye

Summary

A new methodology for constructing convex optimization models called disciplined convex programming is introduced. The methodology enforces a set of conventions upon the models constructed, in turn allowing much of the work required to analyze and solve the models to be automated.

Key words

Convex programming automatic verification symbolic computation modelling language 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, and D. Sorensen. LAPACK Users’ Guide. SIAM, 1992.Google Scholar
  2. 2.
    W. Achtziger, M. Bendsoe, A. Ben-Tal, and J. Zowe. Equivalent displacement based formulations for maximum strength truss topology design. Impact of Computing in Science and Engineering, 4(4):315–45, December 1992.MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    M. Avriel, R. Dembo, and U. Passy. Solution of generalized geometric programs. International Journal for Numerical Methods in Engineering, 9:149–168, 1975.MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    M. Abdi, H. El Nahas, A. Jard, and E. Moulines. Semidefinite positive relaxation of the maximum-likelihood criterion applied to multiuser detection in a CDMA context. IEEE Signal Processing Letters, 9(6):165–167, June 2002.CrossRefGoogle Scholar
  5. 5.
    F. Alizadeh. Interior point methods in semidefinite programming with applications to combinatorial optimization. SIAM Journal on Optimization, 5(1):13–51, February 1995.MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    B. Alkire and L. Vandenberghe. Convex optimization problems involving finite autocorrelation sequences. Mathematical Programming, Series A, 93:331–359, 2002.MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    E. Andersen and Y. Ye. On a homogeneous algorithm for the monotone complementarity problem. Mathematical Programming, 84:375–400, 1999.MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    S. Boyd and C. Barratt. Linear Controller Design: Limits of Performance. Prentice-Hall, 1991.Google Scholar
  9. 9.
    M. Bendsoe, A. Ben-Tal, and J. Zowe. Optimization methods for truss geometry and topology design. Structural Optimization, 7:141–159, 1994.CrossRefGoogle Scholar
  10. 10.
    C. Bischof, A. Carle, G. Corliss, A. Grienwank, and P. Hovland. ADIFOR: Generating derivative codes from Fortran programs. Scientific Programming, pages 1–29, December 1991.Google Scholar
  11. 11.
    S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan. Linear Matrix Inequalities in System and Control Theory. SIAM, 1994.Google Scholar
  12. 12.
    S. Benson. DSDP 4.5: A daul scaling algorithm for semidefinite programming. Web site: http://www-unix.mcs.anl.gov/~benson/dsdp/, March 2002.Google Scholar
  13. 13.
    D. Bertsekas. Nonlinear Programming. Athena Scientific, Belmont, Massachusetts, 1995.MATHGoogle Scholar
  14. 14.
    R. Byrd, N. Gould, J. Norcedal, and R. Waltz. An active-set algorithm for nonlinear programming using linear programming and equality constrained subproblems. Technical Report OTC 2002/4, Optimization Technology Center, Northwestern University, October 2002.Google Scholar
  15. 15.
    O. Bahn, J. Goffin, J. Vial, and O. Du Merle. Implementation and behavior of an interior point cutting plane algorithm for convex programming: An application to geometric programming. Working Paper, University of Geneva, Geneva, Switzerland, 1991.Google Scholar
  16. 16.
    S. Boyd, M. Hershenson, and T. Lee. Optimal analog circuit design via geometric programming, 1997. Preliminary Patent Filing, Stanford Docket S97–122.Google Scholar
  17. 17.
    R. Banavar and A. Kalele. A mixed norm performance measure for the design of multirate filterbanks. IEEE Transactions on Signal Processing, 49(2):354–359, February 2001.CrossRefGoogle Scholar
  18. 18.
    A. Brooke, D. Kendrick, A. Meeraus, and R. Raman. GAMS: A User’s Guide. The Scientific Press, South San Francisco, 1998. Web site: http://www.gams.com/docs/gams/GAMSUsersGuide.pdf.Google Scholar
  19. 19.
    J. Borwein and A. Lewis. Duality relationships for entropy-like minimization problems. SIAM J. Control and Optimization, 29(2):325–338, March 1991.MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    D. Bertsimas and J. Nino-Mora. Optimization of multiclass queuing networks with changeover times via the achievable region approach: part ii, the multistation case. Mathematics of Operations Research, 24(2), May 1999.Google Scholar
  21. 21.
    D. Bertsekas, A. Nedic, and A. Ozdaglar. Convex Analysis and Optimization. Athena Scientific, Nashua, New Hampshire, 2004.Google Scholar
  22. 22.
    B. Borchers. CDSP, a C library for semidefinite programming. Optimization Methods and Software, 11:613–623, 1999.MATHMathSciNetGoogle Scholar
  23. 23.
    A. Ben-Tal and M. Bendsoe. A new method for optimal truss topology design. SIAM J. Optim., 13(2), 1993.Google Scholar
  24. 24.
    A. Ben-Tal, M. Kocvara, A. Nemirovski, and J. Zowe. Free material optimization via semidefinite programming: the multiload case with contact conditions. SIAM Review, 42(4):695–715, 2000.MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    A. Ben-Tal and A. Nemirovski. Interior point polynomial time method for truss topology design. SIAM Journal on Optimization, 4(3):596–612, August 1994.MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    A. Ben-Tal and A. Nemirovski. Robust truss topology design via semidefinite programming. SIAM J. Optim., 7(4):991–1016, 1997.MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    A. Ben-Tal and A. Nemirovski. Structural design via semidefinite programming. In Handbook on Semidefinite Programming, pages 443–467. Kluwer, Boston, 2000.Google Scholar
  28. 28.
    S. Boyd and L. Vandenberghe. Semidefinite programming relaxations of non-convex problems in control and combinatorial optimization. In A. Paulraj, V. Roychowdhuri, and C. Schaper, editors, Communications, Computation, Control and Signal Processing: a Tribute to Thomas Kailath, chapter 15, pages 279–288. Kluwer Academic Publishers, 1997.Google Scholar
  29. 29.
    S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press, 2004.Google Scholar
  30. 30.
    P. Biswas and Y. Ye. Semidefinite programming for ad hoc wireless sensor network localization. Technical report, Stanford University, April 2004. Web site: http: //www.stanford.edu/~yyye/adhocn4.pdf.Google Scholar
  31. 31.
    A. Conn, N. Gould, D. Orban, and Ph. Toint. A primal-dual trust-region algorithm for non-convex nonlinear programming. Mathematical Programming, 87:215–249, 2000.MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    A. Conn, N. Gould, and Ph. Toint. LANCELOT: a Fortran Package for Large-Scale Nonlinear Optimization (Release A), volume 17 of Springer Series in Computational Mathematics. Springer Verlag, 1992.Google Scholar
  33. 33.
    A. Conn, N. Gould, and Ph. Toint. Trust-Region Methods. Series on Optimization. SIAM/MPS, Philadelphia, 2000.MATHGoogle Scholar
  34. 34.
    J. Chinneck. MProbe 5.0 (software package). Web site: http://www.sce.carleton.ca/facuity/chinneck/mprobe.html, December 2003.Google Scholar
  35. 35.
    G. Calafiore and M. Indri. Robust calibration and control of robotic manipulators. In American Control Conference, pages 2003–2007, 2000.Google Scholar
  36. 36.
    C. Crusius. A parser/solver for convex optimization problems. PhD thesis, Stanford University, 2002.Google Scholar
  37. 37.
    T. Terlaky C. Roos and J.-Ph. Vial. Interior Point Approach to Linear Optimization: Theory and Algorithms. John Wiley & Sons, New York, NY, 1997.Google Scholar
  38. 38.
    G. B. Dantzig. Linear Programming and Extensions. Princeton University Press, 1963.Google Scholar
  39. 39.
    J. Dawson, S. Boyd, M. Hershenson, and T. Lee. Optimal allocation of local feedback in multistage amplifiers via geometric programming. IEEE Journal of Circuits and Systems I, 48(1):1–11, January 2001.CrossRefGoogle Scholar
  40. 40.
    M. Dahleh and I. Diaz-Bobillo. Control of Uncertain Systems. A Linear Programming Approach. Prentice Hall, 1995.Google Scholar
  41. 41.
    S. Dirkse and M. Ferris. The PATH solver: A non-monotone stabilization scheme for mixed complementarity problems. Optimization Methods and Software, 5:123–156, 1995.Google Scholar
  42. 42.
    Y. Doids, V. Guruswami, and S. Khanna. The 2-catalog segmentation problem. In Proceedings of SODA, pages 378–380, 1999.Google Scholar
  43. 43.
    T. Davidson, Z. Luo, and K. Wong. Design of orthogonal pulse shapes for communications via semidefinite programming. IEEE Transactions on Communications, 48(5):1433–1445, May 2000.Google Scholar
  44. 44.
    G. Dullerud and F. Paganini. A Course in Robust Control Theory, volume 36 of Texts in Applied Mathematics. Springer-Verlag, 2000.Google Scholar
  45. 45.
    C. de Souza, R. Palhares, and P. Peres. Robust H filter design for uncertain linear systems with multiple time-varying state delays. IEEE Transactions on Signal Processing, 49(3):569–575, March 2001.MathSciNetCrossRefGoogle Scholar
  46. 46.
    A. Doherty, P. Parrilo, and F. Spedalieri. Distinguishing separable and entangled states. Physical Review Letters, 88(18), 2002.Google Scholar
  47. 47.
    B. Dumitrescu, I. Tabus, and P. Stoica. On the parameterization of positive real sequences and MA parameter estimation. IEEE Transactions on Signal Processing, 49(11):2630–2639, November 2001.MathSciNetCrossRefGoogle Scholar
  48. 48.
    R. Duffin. Linearizing geometric programs. SIAM Review, 12:211–227, 1970.MATHMathSciNetCrossRefGoogle Scholar
  49. 49.
    C. Du, L. Xie, and Y. Soh. H filtering of 2-D discrete systems. IEEE Transactions on Signal Processing, 48(6): 1760–1768, June 2000.CrossRefMATHGoogle Scholar
  50. 50.
    H. Du, L. Xie, and Y. Soh. H reduced-order approximation of 2-D digital filters. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 48(6):688–698, June 2001.CrossRefMATHGoogle Scholar
  51. 51.
    Laurent El Ghaoui, Jean-Luc Commeau, Francois Delebecque, and Ramine Nikoukhah. LMITOOL 2.1 (software package). Web site: http://robotics.eecs.berkeley.edu/~elghaoui/lmitool/lmitool.html, March 1999.Google Scholar
  52. 52.
    J. Ecker. Geometric programming: methods, computations and applications. SIAM Rev., 22(3):338–362, 1980.MATHMathSciNetCrossRefGoogle Scholar
  53. 53.
    J.-P. A. Haeberly F. Alizadeh and M. Overton. Primal-dual interior-point methods for semidefinite programming: Convergence rates, stability and numerical results. SIAM J. Optimization, 8:46–76, 1998.MathSciNetGoogle Scholar
  54. 54.
    M. Fu, C. de Souza, and Z. Luo. Finite-horizon robust Kalman filter design. IEEE Transactions on Signal Processing, 49(9):2103–2112, September 2001.MathSciNetCrossRefGoogle Scholar
  55. 55.
    U. Feige and M. Goemans. Approximating the value of two prover proof systems, with applications to max 2sat and max dicut. In Proceedings of the 3nd Israel Symposium on Theory and Computing Systems, pages 182–189, 1995.Google Scholar
  56. 56.
    R. Fourer, D. Gay, and B. Kernighan. AMPL: A Modeling Language for Mathematical Programming. Duxbury Press, December 1999.Google Scholar
  57. 57.
    A. Frieze and M. Jerrum. Improved approximation algorithms for max k-cut and max bisection. Algorithmica, 18:67–81, 1997.MathSciNetMATHGoogle Scholar
  58. 58.
    K. Fujisawa, M. Kojima, K. Nakata, and M. Yamashita. SDPA (Semi-Definite Programming Algorithm) user’s manual—version 6.00. Technical report, Tokyo Insitute of Technology, July 2002.Google Scholar
  59. 59.
    U. Feige and M. Langberg. Approximation algorithms for maximization problems arising in graph partitioning. Journal of Algorithms, 41:174–211, 2001.MathSciNetCrossRefMATHGoogle Scholar
  60. 60.
    U. Feige and M. Langberg. The rpr 2 rounding technique for semidefinte programs. In ICALP, Lecture Notes in Computer Science. Springer, Berlin, 2001.Google Scholar
  61. 61.
    R. Fourer. Nonlinear programming frequently asked questions. Web site: http://www-unix.mcs.anl.gov/otc/Guide/faq/nonlinear-programming%-iaq.html, 2000.Google Scholar
  62. 62.
    R. Freund. Polynomial-time algorithms for linear programming based only on primal scaling and projected gradients of a potential function. Mathematical Programming, 51:203–222, 1991.MATHMathSciNetCrossRefGoogle Scholar
  63. 63.
    Frontline Systems, Inc. Premium Solver Platform (software package). Web site: http://www.solver.com, September 2004.Google Scholar
  64. 64.
    E. Fridman and U. Shaked. A new H filter design for linear time delay systems. IEEE Transactions on Signal Processing, 49(11):2839–2843, July 2001.MathSciNetCrossRefGoogle Scholar
  65. 65.
    J. Geromel. Optimal linear filtering under parameter uncertainty. IEEE Transactions on Signal Processing, 47(1):168–175, January 1999.MATHMathSciNetCrossRefGoogle Scholar
  66. 66.
    O. Güler and R. Hauser. Self-scaled barrier functions on symmetric cones and their classification. Foundations of Computational Mathematics, 2:121–143, 2002.MathSciNetMATHGoogle Scholar
  67. 67.
    D. Goldfarb and G. Iyengar. Robust portfolio selection problems. Technical report, Computational Optimization Research Center, Columbia University, March 2002. Web site: http://www.corc.ieor.columbia.edu/reports/techreports/tr-2002-03.pdf.Google Scholar
  68. 68.
    D. Goldfarb and G. Iyengar. Robust quadratically constrained problems program. Technical Report TR-2002-04, Department of IEOR, Columbia University, New York, NY USA, 2002.Google Scholar
  69. 69.
    P. Gill, W. Murray, and M. Saunders. SNOPT: An sqp algorithm for large-scale constrained optimization. SIAM Journal on Optimization, 12:979–1006, 2002.MathSciNetCrossRefMATHGoogle Scholar
  70. 70.
    P. Gill, W. Murray, M. Saunders, and M. Wright. User’s guide for NPSOL 5.0: A FORTRAN package for nonlinear programming. Technical Report SOL 86-1, Systems Optimization Laboratory, Stanford University, July 1998. Web site: http://www.sbsi-sol-optimize.com/manuals/NPSOL%205-0%20Manual.p%df.Google Scholar
  71. 71.
    P. Gill, W. Murray, and M. Wright. Practical Optimization. Academic Press, London, 1981.MATHGoogle Scholar
  72. 72.
    J. Geromel and M. De Oliveira. H 2/H robust filtering for convex bounded uncertain systems. IEEE Transactions on Automatic Control, 46(1):100–107, January 2001.CrossRefMATHGoogle Scholar
  73. 73.
    C. Gonzaga. Path following methods for linear programming. SIAM Review, 34(2):167–227, 1992.MATHMathSciNetCrossRefGoogle Scholar
  74. 74.
    M. Grant. Disciplined Convex Programming. PhD thesis, Department of Electrical Engineering, Stanford University, December 2004.Google Scholar
  75. 75.
    D. Guo, L. Rasmussen, S. Sun, and T. Lim. A matrix-algebraic approach to linear parallel interference cancellation in CDMA. IEEE Transactions on Communications, 48(1):152–161, January 2000.CrossRefGoogle Scholar
  76. 76.
    G. Golub and C. Van Loan. Matrix Computations. Johns Hopkins Univ. Press, Baltimore, second edition, 1989.MATHGoogle Scholar
  77. 77.
    M. Goemans and D. Williamson. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of the ACM, 42:1115–1145, 1995.MathSciNetCrossRefMATHGoogle Scholar
  78. 78.
    M. Hershenson, S. Boyd, and T. Lee. Optimal design of a CMOS op-amp via geometric programming. IEEE Transactions on Computer-Aided Design, January 2001.Google Scholar
  79. 79.
    L. Huaizhong and M. Fu. A linear matrix inequality approach to robust H filtering. IEEE Transactions on Signal Processing, 45(9):2338–2350, September 1997.CrossRefGoogle Scholar
  80. 80.
    M. Hershenson, S. Mohan, S. Boyd, and T. Lee. Optimization of inductor circuits via geometric programming. In Proceedings 36th IEEE/ACM Integrated Circuit Design Automation Conference, 1999.Google Scholar
  81. 81.
    P. Hovland, B. Norris, and C. Bischof. ADIC (software package), November 2003. http://www-fp.mcs.anl.gov/adic/.Google Scholar
  82. 82.
    L. Han, J. Trinkle, and Z. Li. Grasp analysis as linear matrix inequality problems. IEEE Transactions on Robotics and Automation, 16(6):663–674, December 2000.CrossRefGoogle Scholar
  83. 83.
    J.-B. Hiriart-Urruty and C. Lemaréchal. Convex Analysis and Minimization Algorithms I, volume 305 of Grundlehren der mathematischen Wissenschaften. Springer-Verlag, New York, 1993.Google Scholar
  84. 84.
    J.-B. Hiriart-Urruty and C. Lemaréchal. Convex Analysis and Minimization Algorithms II: Advanced Theory and Bundle Methods, volume 306 of Grundlehren der mathematischen Wissenschaften. Springer-Verlag, New York, 1993.MATHGoogle Scholar
  85. 85.
    Q. Han, Y. Ye, and J. Zhang. An improved rounding method and semidefinite programming relaxation for graph partition. Math. Programming, 92:509–535, 2002.MathSciNetCrossRefMATHGoogle Scholar
  86. 86.
    F. Jarre, M. Kocvara, and J. Zowe. Optimal truss design by interior point methods. SIAM J. Optim., 8(4):1084–1107, 1998.MathSciNetCrossRefMATHGoogle Scholar
  87. 87.
    F. Jarre and M. Saunders. A practical interior-point method for convex programming. SIAM Journal on Optimization, 5:149–171, 1995.MathSciNetCrossRefMATHGoogle Scholar
  88. 88.
    N. Karmarkar. A new polynomial-time algorithm for linear programming. Combinatorica, 4(4):373–395, 1984.MATHMathSciNetGoogle Scholar
  89. 89.
    M. Kojima, S. Mizuno, and A. Yoshise. An O(√nL)-iteration potential reduction algorithm for linear complementarity problems. Mathematical Programming, 50:331–342, 1991.MathSciNetCrossRefMATHGoogle Scholar
  90. 90.
    J. Kleinberg, C. Papadimitriou, and P. Raghavan. Segmentation problems. In Proceedings of the 30th Symposium on Theory of Computation, pages 473–482, 1998.Google Scholar
  91. 91.
    J. Keuchel, C. Schnörr, C. Schellewald, and D. Cremers. Binary partitioning, perceptual grouping, and restoration with semidefinite programming. IEEE Transactions on Pattern Analysis and Machine Intelligence, 25(11):1364–1379, November 2003.CrossRefGoogle Scholar
  92. 92.
    K. Kortanek, X. Xu, and Y. Ye. An infeasible interior-point algorithm for solving primal and dual geometric progams. Mathematical Programming, 1:155–181, 1997.MathSciNetCrossRefGoogle Scholar
  93. 93.
    K. Kortanek, X. Xu, and Y. Ye. An infeasible interior-point algorithm for solving primal and dual geometric programs. Mathematical Programming, 76:155–182, 1997.MathSciNetCrossRefGoogle Scholar
  94. 94.
    M. Kocvara, J. Zowe, and A. Nemirovski. Cascading-an approach to robust material optimization. Computers and Structures, 76:431–442, 2000.CrossRefGoogle Scholar
  95. 95.
    J. Lasserre. Global optimization with polynomials and the problem of moments. SIAM Journal of Optimization, 11:796–817, 2001.MATHMathSciNetCrossRefGoogle Scholar
  96. 96.
    J. Lasserre. Bounds on measures satisfying moment conditions. Annals of Applied Probability, 12:1114–1137, 2002.MATHMathSciNetCrossRefGoogle Scholar
  97. 97.
    J. Lasserre. Semidefinite programming vs. LP relaxation for polynomial programming. Mathematics of Operations Research, 27(2):347–360, May 2002.MATHMathSciNetCrossRefGoogle Scholar
  98. 98.
    H. Lebret and S. Boyd. Antenna array pattern synthesis via convex optimization. IEEE Transactions on Signal Processing, 45(3):526–532, March 1997.CrossRefGoogle Scholar
  99. 99.
    Lindo Systems, Inc. LINGO version 8.0 (software package). Web site: http: //www.lindo.com, September 2004.Google Scholar
  100. 100.
    J. Löfberg. YALMIP verison 2.1 (software package). Web site: http://www.control.isy.liu.se/~johanl/yalmip.html, September 2001.Google Scholar
  101. 101.
    L. Lovasz. An Algorithmic Theory of Numbers, Graphs and Convexity, volume 50 of CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1986.Google Scholar
  102. 102.
    Z.-Q. Luo, J. Sturm, and S. Zhang. Duality and self-duality for conic con-vex programming. Technical report, Department of Electrical and Computer Engineering, McMaster University, 1996.Google Scholar
  103. 103.
    W. Lu. A unified approach for the design of 2-D digital filters via semidefinite programming. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 49(6):814–826, June 2002.MathSciNetCrossRefGoogle Scholar
  104. 104.
    M. Lobo, L. Vandenberghe, S. Boyd, and H. Lebret. Applications of second-order cone programming. Linear Algebra and its Applications, 284:193–228, November 1998. Special issue on Signals and Image Processing.MathSciNetCrossRefMATHGoogle Scholar
  105. 105.
    R. Monteiro and I. Adler. Interior path following primal-dual algorithms: Part I: Linear programming. Mathematical Programming, 44:27–41, 1989.MathSciNetCrossRefMATHGoogle Scholar
  106. 106.
    The MathWorks, Inc. PRO-MATLAB User’s Guide. The MathWorks, Inc., 1990.Google Scholar
  107. 107.
    M. Mahmoud and A. Boujarwah. Robust H filtering for a class of linear parameter-varying systems. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 48(9):1131–1138, September 2001.MathSciNetCrossRefMATHGoogle Scholar
  108. 108.
    G. Millerioux and J. Daafouz. Global chaos synchronization and robust filtering in noisy context. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 48(10):1170–1176, October 2001.MathSciNetCrossRefGoogle Scholar
  109. 109.
    N. Megiddo. Pathways to the optimal set in linear programming. In N. Megiddo, editor, Progress in Mathematical Programming: Interior Point and Related Methods, pages 131–158. Springer Verlag, New York, 1989. Identical version in: Proceedings of the 6th Mathematical Programming Symposium of Japan, Nagoya, Japan, 1–35, 1986.Google Scholar
  110. 110.
    MOSEK ApS. Mosek (software package). Web site: http://www.mosek.com, July 2001.Google Scholar
  111. 111.
    S. Mahajan and H. Ramesh. Derandomizing semidefinite programming based approximation algorithms. SIAM J. of Computing, 28:1641–1663, 1999.MathSciNetCrossRefMATHGoogle Scholar
  112. 112.
    B. Murtaugh and M. Saunders. MINOS 5.5 user’s guide. Technical report, Systems Optimizaiton Laboratory, Stanford University, July 1998. Web site: http://www.sbsi-sol-optimize.com/manuals/Minos%205-5%20Manual.p%df.Google Scholar
  113. 113.
    J. Moré and D. Sorensen. NMTR (software package), March 2000. Web site: http://www-unix.mcs.anl.gov/~more/nmtr/.Google Scholar
  114. 114.
    M. Milanese and A. Vicino. Optimal estimation theory for dynamic systems with set membership uncertainty: An overview. Automatica, 27(6):997–1009, November 1991.MathSciNetCrossRefMATHGoogle Scholar
  115. 115.
    Y. Nesterov. Introductory Lectures on Convex Optimization: A Basic Course, volume 87 of Applied Optimization. Kluwer, Boston, 2004.MATHGoogle Scholar
  116. 116.
    I. Nenov, D. Fylstra, and L. Kolev. Convexity determination in the microsoft excel solver using automatic differentiation techniques. In The 4th Internation Conference on Automatic Differentiation, 2004.Google Scholar
  117. 117.
    Yu. Nesterov and A. Nemirovsky. A general approach to polynomial-time algorithms design for convex programming. Technical report, Centr. Econ. & Math. Inst., USSR Acad. Sci., Moscow, USSR, 1988.Google Scholar
  118. 118.
    Yu. Nesterov and A. Nemirovsky. Interior-Point Polynomial Algorithms in Convex Programming: Theory and Algorithms, volume 13 of Studies in Applied Mathematics. Society of Industrial and Applied Mathematics (SIAM) Publications, Philadelphia, PA 19101, USA, 1993.Google Scholar
  119. 119.
    Yu. Nesterov, O. Pèton, and J.-Ph. Vial. Homogeneous analytic center cutting plane methods with approximate centers. In F. Potra, C. Roos, and T. Terlaky, editors, Optimization Methods and Software, pages 243–273, November 1999. Special Issue on Interior Point Methods.Google Scholar
  120. 120.
    S. Nash and A. Sofer. A barrier method for large-scale constrained optimization. ORSA Journal on Computing, 5:40–53, 1993.MathSciNetMATHGoogle Scholar
  121. 121.
    Yu. Nesterov and M. Todd. Self-scaled barriers and interior-point methods for convex programming. Mathematics of Operations Research, 22:1–42, 1997.MathSciNetCrossRefMATHGoogle Scholar
  122. 122.
    J. Nocedal and S. Wright. Numerical Optimization. Springer Series in Operations Research. Springer, New York, 1999.MATHGoogle Scholar
  123. 123.
    D. Orban and R. Fourer. DrAmpl: a meta-solver for optimization. Technical report, Ecole Poytechnique de Montreal, 2004.Google Scholar
  124. 124.
    M. Overton and R. Womersley. On the sum of the largest eigenvalues of a symmetric matrix. SIAM Journal on Matrix Analysis and Applications, 13(1):41–45, January 1992.MathSciNetCrossRefMATHGoogle Scholar
  125. 125.
    P. Parrilo. Semidefinite programming relaxations for semialgebraic problems. Mathematical Programming, Series B, 96(2):293–320, 2003.MATHMathSciNetCrossRefGoogle Scholar
  126. 126.
    G. Pataki. Geometry of cone-optimization problems and semi-definite programs. Technical report, GSIA Carnegie Mellon University, Pittsburgh, PA, 1994.Google Scholar
  127. 127.
    J. Park, H. Cho, and D. Park. Design of GBSB neural associative memories using semidefinite programming. IEEE Transactions on Neural Networks, 10(4):946–950, July 1999.CrossRefGoogle Scholar
  128. 128.
    R. Palhares, C. de Souza, and P. Dias Peres. Robust H filtering for uncertain discrete-time state-delayed systems. IEEE Transactions on Signal Processing, 48(8):1696–1703, August 2001.CrossRefGoogle Scholar
  129. 129.
    R. Palhares and P. Peres. LMI approach to the mixed H 2/H filtering design for discrete-time uncertain systems. IEEE Transactions on Aerospace and Electronic Systems, 37(1):292–296, January 2001.CrossRefGoogle Scholar
  130. 130.
    S. Prajna, A. Papachristodoulou, and P. Parrilo. SOSTOOLS: Sum of squares optimization toolbox for MATLAB, 2002. Available from http://www.cds.caltech.edu/sostools and http://www.aut.ee.ethz.ch/parrilo/sostools.Google Scholar
  131. 131.
    O. Pèton and J.-P. Vial. A tutorial on ACCPM: User’s guide for version 2.01. Technical Report 2000.5, HEC/Logilab, University of Geneva, March 2001. See also the http://ecolu-info.unige. ch/~logilab/software/accpm/accpm.html.Google Scholar
  132. 132.
    E. Rimon and S. Boyd. Obstacle collision detection using best ellipsoid fit. Journal of Intelligent and Robotic Systems, 18:105–126, 1997.CrossRefMATHGoogle Scholar
  133. 133.
    J. Renegar. A polynomial-time algorithm, based on Newton’s method, for linear programming. Mathematical Programming, 40:59–93, 1988.MATHMathSciNetCrossRefGoogle Scholar
  134. 134.
    B. Radig and S. Florczyk. Evaluation of Convex Optimization Techniques for the Weighted Graph-Matching Problem in Computer Vision, pages 361–368. Springer, December 2001.Google Scholar
  135. 135.
    L. Rasmussen, T. Lim, and A. Johansson. A matrix-algebraic approach to successive interference cancellation in CDMA. IEEE Transactions on Communications, 48(1):145–151, January 2000.CrossRefGoogle Scholar
  136. 136.
    M. Rijckaert and X. Martens. Analysis and optimization of the williams-otto process by geometric programming. AIChE Journal, 20(4):742–750, July 1974.CrossRefGoogle Scholar
  137. 137.
    R. Rockafellar. Convex Analysis. Princeton Univ. Press, Princeton, New Jersey, second edition, 1970.MATHGoogle Scholar
  138. 138.
    E. Rosenberg. Globally Convergent Algorithms for Convex Programming with Applications to Geometric Programming. PhD thesis, Department of Operations Research, Stanford University, 1979.Google Scholar
  139. 139.
    P. Stoica, T. McKelvey, and J. Mari. Ma estimation in polynomial time. IEEE Transactions on Signal Processing, 48(7): 1999–2012, July 2000.MathSciNetCrossRefMATHGoogle Scholar
  140. 140.
    J. Sturm. Using SeDuMi 1.02, a MATLAB toolbox for optimization over sym-metric cones. Optimization Methods and Software, 11:625–653, 1999.MATHMathSciNetGoogle Scholar
  141. 141.
    J. A. K. Suykens, T. Van Gestel, J. De Brabanter, B. De Moor, and J. Vande-walle. Least squares support vector machines, 2002.Google Scholar
  142. 142.
    J. Stoer and C. Witzgall. Convexity and Optimization in Finite Dimensions I. Springer-Verlag, 1970.Google Scholar
  143. 143.
    U. Shaked, L. Xie, and Y. Soh. New approaches to robust minimum variance filter design. IEEE Transactions on Signal Processing, 49(11):2620–2629, November 2001.MathSciNetCrossRefGoogle Scholar
  144. 144.
    H. Tuan, P. Apkarian, and T. Nguyen. Robust and reduced-order filtering: new LMI-based characterizations and methods. IEEE Transactions on Signal Processing, 49(12):2975–2984, December 2001.CrossRefGoogle Scholar
  145. 145.
    H. Tuan, P. Apkarian, T. Nguyen, and T. Narikiyo. Robust mixed H 2/H filtering of 2-D systems. IEEE Transactions on Signal Processing, 50(7):1759–1771, July 2002.CrossRefGoogle Scholar
  146. 146.
    C. Tseng and B. Chen. H , fuzzy estimation for a class of nonlinear discrete-time dynamic systems. IEEE Transactions on Signal Processing, 49(11):2605–2619, November 2001.CrossRefGoogle Scholar
  147. 147.
    The Mathworks, Inc. LMI control toolbox 1.0.8 (software package). Web site: http://www.mathworks.com/products/lmi, August 2002.Google Scholar
  148. 148.
    H. Tan and L. Rasmussen. The application of semidefinite programming for detection in CDMA. IEEE Journal on Selected Areas in Communications, 19(8):1442–1449, August 2001.CrossRefGoogle Scholar
  149. 149.
    T. Tsuchiya. A polynomial primal-dual path-following algorithm for second-order cone programming. Technical report, The Institute of Statistical Mathematics, Tokyo, Japan, October 1997.Google Scholar
  150. 150.
    Z. Tan, Y. Soh, and L. Xie. Envelope-constrained H filter design: an LMI optimization approach. IEEE Transactions on Signal Processing, 48(10):2960–2963, October 2000.MathSciNetCrossRefMATHGoogle Scholar
  151. 151.
    Z. Tan, Y. Soh, and L. Xie. Envelope-constrained H FIR filter design. IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, 47(1):79–82, January 2000.CrossRefMATHGoogle Scholar
  152. 152.
    R. Tütüncü, K. Toh, and M. Todd. SDPT3—a MATLAB software package for semidefinite-quadratic-linear programming, version 3.0. Technical report, Carnegie Mello University, August 2001.Google Scholar
  153. 153.
    R. Tapia, Y. Zhang, and L. Velazquez. On convergence of minimization methods: Attraction, repulsion and selection. Journal of Optimization Theory and Applications, 107:529–546, 2000.MathSciNetCrossRefMATHGoogle Scholar
  154. 154.
    R. Vanderbei. LOQO user’s manual—version 4.05. Technical report, Operations Research and Financial Engineering, Princeton University, October 2000.Google Scholar
  155. 155.
    L. Vandenberghe and S. Boyd. Semidefinite programming. SIAM Review, 38(1):49–95, March 1996.MathSciNetCrossRefMATHGoogle Scholar
  156. 156.
    R. Vanderbei and H. Benson. On forumulating semidefinite programming problems as smooth convex nonlinear optimization problems. Technical Report ORFE-99-01, Operations Research and Financial Engineering, Princeton University, January 2000.Google Scholar
  157. 157.
    L. Vandenberghe, S. Boyd, and A. El Gamal. Optimal wire and transistor sizing for circuits with non-tree topology. In Proceedings of the 1997 IEEE/ACM International Conference on Computer Aided Design, pages 252–259, 1997.Google Scholar
  158. 158.
    L. Vandenberghe, S. Boyd, and A. El Gamal. Optimizing dominant time constant in RC circuits. IEEE Transactions on Computer-Aided Design, 2(2): 110–125, February 1998.CrossRefGoogle Scholar
  159. 159.
    S.-P. Wu and S. Boyd. SDPSOL: A parser/solver for semidefinite programs with matrix structure. In L. El Ghaoui and S.-I. Niculescu, editors, Recent Advances in LMI Methods for Control, chapter 4, pages 79–91. SIAM, 2000.Google Scholar
  160. 160.
    F. Wang and V. Balakrishnan. Robust Kalman filters for linear time-varying systems with stochastic parametric uncertainties. IEEE Transactions on Signal Processing, 50(4):803–813, April 2002.MathSciNetCrossRefGoogle Scholar
  161. 161.
    S.-P. Wu, S. Boyd, and L. Vandenberghe. FIR filter design via spectral factorization and convex optimization. In B. Datta, editor, Applied and Computational Control, Signals, and Circuits, volume 1, pages 215–245. Birkhauser, 1998.Google Scholar
  162. 162.
    M. Wright. Some properties of the Hessian of the logarithmic barrier function. Mathematical Programming, 67:265–295, 1994.MATHMathSciNetCrossRefGoogle Scholar
  163. 163.
    S. Wright. Primal Dual Interior Point Methods. Society of Industrial and Applied Mathematics (SIAM) Publications, Philadelphia, PA 19101, USA, 1999.Google Scholar
  164. 164.
    J. Weickert and Christoph Schnörr. A theoretical framework for convex regularizers in pde-based computation of image motion. International Journal of Computer Vision, Band 45, 3:245–264, 2001.CrossRefGoogle Scholar
  165. 165.
    S. Wang, L. Xie, and C. Zhang. H2 optimal inverse of periodic FIR digital filters. IEEE Transactions on Signal Processing, 48(9):2696–2700, September 2000.CrossRefGoogle Scholar
  166. 166.
    Y. Ye. Interior-point algorithms: Theory and practice. John Wiley & Sons, New York, NY, 1997.Google Scholar
  167. 167.
    Y. Ye. A path to the arrow-debreu competitive market equilibrium. Technical report, Stanford University, February 2004. Web site: http://www.stanford. edu/~yyye/arrow-debreu2.pdf.Google Scholar
  168. 168.
    F. Yang and Y. Hung. Robust mixed H 2/H filtering with regional pole assignment for uncertain discrete-time systems. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 49(8): 1236–1241, August 2002.MathSciNetCrossRefGoogle Scholar
  169. 169.
    Y. Ye, M. Todd, and S. Mizuno. An O(√nL)-iteration homogeneous and self-dual linear programming algorithm. Mathematics of Operations Research, 19(1):53–67, 1994.MathSciNetMATHGoogle Scholar
  170. 170.
    Y. Ye and J. Zhang. Approximation for dense-n/2-subgraph and the complement of min-bisection. Manuscript, 1999.Google Scholar
  171. 171.
    S. Zhang. A new self-dual embedding method for convex programming. Technical report, Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, October 2001.Google Scholar
  172. 172.
    M. Zibulevsky. Pattern recognition via support vector machine with computationally efficient nonlinear transform. Technical report, The University of New Mexico, Computer Science Department, 1998. Web site: http: //iew3.technion.ac.il/~mcib/nipspsvm.ps.gz.Google Scholar
  173. 173.
    J. Zowe, M. Kocvara, and M. Bendsoe. Free material optimization via mathematical programming. Mathematical Programming, 9:445–466, 1997.MathSciNetCrossRefGoogle Scholar
  174. 174.
    U. Zwick. Outward rotations: a tool for rounding solutions of semidefinite programming relaxations, with applications to max cut and other problems. In Proceedings of the 31th Symposium on Theory of Computation, pages 679–687, 1999.Google Scholar
  175. 175.
    H. Zhou, L. Xie, and C. Zhang. A direct approach to H 2 optimal deconvolution of periodic digital channels. IEEE Transactions on Signal Processing, 50(7):1685–1698, July 2002.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • Michael Grant
    • 1
  • Stephen Boyd
    • 1
  • Yinyu Ye
    • 1
    • 2
  1. 1.Department of Electrical EngineeringStanford UniversityUSA
  2. 2.Department of Management Science and EngineeringStanford UniversityUSA

Personalised recommendations