Algorithms and software for LMI problems in control

A number of important problems from system and control theory can be numerically solved by reformulating them as convex optimization problems with linear matrix inequality (LMI) constraints. While numerous articles have appeared cataloging applications of LMIs to control system analysis and design, there have been few publications in the control literature describing the numerical solution of these optimization problems. The purpose of this article is to provide an overview of the state of the art of numerical algorithms for LMI problems, and of the available software.

[1]  Bùi-Trong-Liêu,et al.  La mèthode des centres dans un espace topologique , 1966 .

[2]  M. J. D. Powell,et al.  Nonlinear Programming—Sequential Unconstrained Minimization Techniques , 1969 .

[3]  A. Hoffman,et al.  Lower bounds for the partitioning of graphs , 1973 .

[4]  P. Wolfe,et al.  The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices , 1975 .

[5]  R. Fletcher.  A Nonlinear Programming Problem in Statistics (Educational Testing) , 1981 .

[6]  Narendra Karmarkar,et al.  A new polynomial-time algorithm for linear programming , 1984, Comb..

[7]  R. Fletcher.  Semi-Definite Matrix Constraints in Optimization , 1985 .

[8]  P. Khargonekar,et al.  State-space solutions to standard H2 and H∞ control problems , 1988, 1988 American Control Conference.

[9]  M. Kojima,et al.  A primal-dual interior point algorithm for linear programming , 1988 .

[10]  M. Overton.  On minimizing the maximum eigenvalue of a symmetric matrix , 1988 .

[11]  P. Khargonekar,et al.  State-space solutions to standard H/sub 2/ and H/sub infinity / control problems , 1989 .

[12]  N. Karmarkar,et al.  A continuous approach to compute upper bounds in quadratic maximization problems with integer constraints , 1992 .

[13]  Margaret H. Wright,et al.  Interior methods for constrained optimization , 1992, Acta Numerica.

[14]  F. Jarre.  Interior-point methods for convex programming , 1992 .

[15]  Michael L. Overton,et al.  Large-Scale Optimization of Eigenvalues , 1990, SIAM J. Optim..

[16]  Roy E. Marsten,et al.  The interior-point method for linear programming , 1992, IEEE Software.

[17]  F. Jarre.  An interior-point method for minimizing the maximum eigenvalue of a linear combination of matrices , 1993 .

[18]  N. Karmarkar,et al.  An O(nL) Iteration Algorithm for Computing Bounds in Quadratic Optimization Problems , 1993 .

[19]  Stephen P. Boyd,et al.  Method of centers for minimizing generalized eigenvalues , 1993, Linear Algebra and its Applications.

[20]  K. Goh,et al.  Control system synthesis via bilinear matrix inequalities , 1994, Proceedings of 1994 American Control Conference - ACC '94.

[21]  G. Papavassilopoulos,et al.  A global optimization approach for the BMI problem , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[22]  K. Goh,et al.  /spl mu//K/sub m/-synthesis via bilinear matrix inequalities , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[23]  L. El Ghaoui,et al.  Synthesis of fixed-structure controllers via numerical optimization , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[24]  G. Papavassilopoulos,et al.  Biaffine matrix inequality properties and computational methods , 1994, Proceedings of 1994 American Control Conference - ACC '94.

[25]  R.E. Skelton.  Increased roles of linear algebra in control education , 1994, Proceedings of 1994 American Control Conference - ACC '94.

[26]  M. Overton,et al.  Optimizing eigenvalues of symmetric definite pencils , 1994, Proceedings of 1994 American Control Conference - ACC '94.

[27]  Leonid Faybusovich.  On a Matrix Generalization of Affine-Scaling Vector Fields , 1995, SIAM J. Matrix Anal. Appl..

[28]  Stephen P. Boyd,et al.  A primal—dual potential reduction method for problems involving matrix inequalities , 1995, Math. Program..

[29]  F. Potra,et al.  Homogeneous Interior{point Algorithms for Semideenite Programming , 1995 .

[30]  Yurii Nesterov,et al.  An interior-point method for generalized linear-fractional programming , 1995, Math. Program..

[31]  Chih-Jen Lin,et al.  A predictor corrector method for semi-definite linear programming , 1995 .

[32]  Shinji Hara,et al.  Interior Point Methods for the Monotone Linear Complementarity Problem in Symmetric Matrices , 1995 .

[33]  Shuzhong Zhang,et al.  Symmetric primal-dual path-following algorithms for semidefinite programming , 1999 .

[34]  Alexander Shapiro,et al.  On Eigenvalue Optimization , 1995, SIAM J. Optim..

[35]  Farid Alizadeh,et al.  Interior Point Methods in Semidefinite Programming with Applications to Combinatorial Optimization , 1995, SIAM J. Optim..

[36]  Venkataramanan Balakrishnan,et al.  Connections between duality in control theory and convex optimization , 1995, Proceedings of 1995 American Control Conference - ACC'95.

[37]  Jacek Gondzio,et al.  Implementation of Interior Point Methods for Large Scale Linear Programming , 1996 .

[38]  L. Vandenberghe,et al.  Algorithms and software tools for LMI problems in control: an overview , 1996, Proceedings of Joint Conference on Control Applications Intelligent Control and Computer Aided Control System Design.

[39]  Shuzhong Zhang,et al.  Polynomial primal-dual cone affine scaling for semidefinite programming , 1999 .

[40]  Robert J. Vanderbei,et al.  An Interior-Point Method for Semidefinite Programming , 1996, SIAM J. Optim..

[41]  Stephen Boyd,et al.  MAXDET: Software for Determinant Maximization Problems User's Guide , 1996 .

[42]  Kim-Chuan Toh,et al.  SDPT3 -- A Matlab Software Package for Semidefinite Programming , 1996 .

[43]  F. Potra,et al.  Superlinear Convergence of a Predictor-corrector Method for Semideenite Programming without Shrinking Central Path Neighborhood , 1996 .

[44]  K. Goh,et al.  Robust synthesis via bilinear matrix inequalities , 1996 .

[45]  E. Beran.  Induced norm control toolbox , 1996, Proceedings of 35th IEEE Conference on Decision and Control.

[46]  Leonid Faybusovich,et al.  Semidefinite Programming: A Path-Following Algorithm for a Linear-Quadratic Functional , 1996, SIAM J. Optim..

[47]  Stephen P. Boyd,et al.  Semidefinite Programming , 1996, SIAM Rev..

[48]  Shuzhong Zhang,et al.  Duality and Self-Duality for Conic Convex Programming , 1996 .

[49]  Michael J. Todd,et al.  Self-Scaled Barriers and Interior-Point Methods for Convex Programming , 1997, Math. Oper. Res..

[50]  Stephen J. Wright.  Primal-Dual Interior-Point Methods , 1997, Other Titles in Applied Mathematics.

[51]  Renato D. C. Monteiro,et al.  Primal-Dual Path-Following Algorithms for Semidefinite Programming , 1997, SIAM J. Optim..

[52]  Arkadi Nemirovski,et al.  The projective method for solving linear matrix inequalities , 1997, Math. Program..

[53]  Henry Wolkowicz,et al.  Strong Duality for Semidefinite Programming , 1997, SIAM J. Optim..

[54]  Arkadi Nemirovski.  The long-step method of analytic centers for fractional problems , 1997, Math. Program..

[55]  Shinji Hara,et al.  Interior-Point Methods for the Monotone Semidefinite Linear Complementarity Problem in Symmetric Matrices , 1997, SIAM J. Optim..

[56]  Michael L. Overton,et al.  Primal-Dual Interior-Point Methods for Semidefinite Programming: Convergence Rates, Stability and Numerical Results , 1998, SIAM J. Optim..

[57]  Michael J. Todd,et al.  Primal-Dual Interior-Point Methods for Self-Scaled Cones , 1998, SIAM J. Optim..

[58]  Shuzhong Zhang,et al.  An interior point method, based on rank-1 updates, for linear programming , 1998, Math. Program..

[59]  Renato D. C. Monteiro,et al.  Polynomial Convergence of Primal-Dual Algorithms for Semidefinite Programming Based on the Monteiro and Zhang Family of Directions , 1998, SIAM J. Optim..

[60]  Masakazu Kojima,et al.  Local convergence of predictor—corrector infeasible-interior-point algorithms for SDPs and SDLCPs , 1998, Math. Program..

[61]  E. Yaz.  Linear Matrix Inequalities In System And Control Theory , 1998, Proceedings of the IEEE.

[62]  Stephen P. Boyd,et al.  Determinant Maximization with Linear Matrix Inequality Constraints , 1998, SIAM J. Matrix Anal. Appl..

[63]  Jianmin Jiang,et al.  A long-step primal-dual path-following method for semidefinite programming , 1998, Oper. Res. Lett..

[64]  Tamás Terlaky,et al.  On primal-dual path-following algorithms in semidefinite programming. , 1998 .

[65]  Yin Zhang,et al.  A unified analysis for a class of long-step primal-dual path-following interior-point algorithms for semidefinite programming , 1998, Math. Program..

[66]  Kim-Chuan Toh,et al.  On the Nesterov-Todd Direction in Semidefinite Programming , 1998, SIAM J. Optim..

[67]  Nathan W. Brixius,et al.  Sdpha: a Matlab implementation of homogeneous interior-point algorithms for semidefinite programming , 1999 .

[68]  Masakazu Kojima,et al.  A Predictor-corrector Interior-point Algorithm for the Semidenite Linear Complementarity Problem Using the Alizadeh-haeberly-overton Search Direction , 1996 .

[69]  B. Borchers.  CSDP, A C library for semidefinite programming , 1999 .

[70]  M. Kojima,et al.  A note on the Nesterov-Todd and the Kojima-Shindoh-hara search directions in semidefinite programming , 1999 .

[71]  Jun Ji,et al.  On the Local Convergence of a Predictor-Corrector Method for Semidefinite Programming , 1999, SIAM J. Optim..

[72]  F. Potra,et al.  On a general class of interior-point algorithms for semidefinite programming with polynomial complexity and superlinear convergence , 1999 .

[73]  Takashi Tsuchiya,et al.  Polynomiality of primal-dual algorithms for semidefinite linear complementarity problems based on the Kojima-Shindoh-Hara family of directions , 1999, Math. Program..