Low-rank structure in semidefinite programs derived from the KYP lemma

We extend a fast technique for solving semidefinite programs involving nonnnegative trigonometric polynomials to problems derived from the discrete-time Kalman-Yakubovich- Popov (KYP) lemma and some of its generalizations. The frequency-domain inequality associated with the generalized KYP lemma is first expressed as a weighted sum of squares of rational functions. By taking a sufficient number of samples of the sum-of-squares expression, an equivalent standard-form semidefinite program with low-rank structure is obtained. This low-rank structure is easily exploited in implementations of primal-dual interior-point algorithms. A complexity analysis and numerical examples are provided to support the performance improvement over standard semidefinite programming solvers.

[1]  Paul Van Dooren,et al.  Optimization Problems over Positive Pseudopolynomial Matrices , 2003, SIAM J. Matrix Anal. Appl..

[2]  Anders Hansson,et al.  Efficient solution of linear matrix inequalities for integral quadratic constraints , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

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

[4]  Alexandre Megretski,et al.  Fast algorithms for solving IQC feasibility and optimization problems , 2001, Proceedings of the 2001 American Control Conference. (Cat. No.01CH37148).

[5]  Bogdan Dumitrescu,et al.  Interior-Point Algorithms for Sum-Of-Squares Optimization of Multidimensional Trigonometric Polynomials , 2007, 2007 IEEE International Conference on Acoustics, Speech and Signal Processing - ICASSP '07.

[6]  Alexandre Megretski,et al.  Specialized fast algorithms for IQC feasibility and optimization problems , 2004, Autom..

[7]  Shinji Hara,et al.  Generalized KYP lemma: unified frequency domain inequalities with design applications , 2005, IEEE Transactions on Automatic Control.

[8]  S. J. Benson,et al.  DSDP5 user guide - software for semidefinite programming. , 2006 .

[9]  P. Parrilo,et al.  From coefficients to samples: a new approach to SOS optimization , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[10]  Yurii Nesterov,et al.  Squared Functional Systems and Optimization Problems , 2000 .

[11]  Masakazu Kojima,et al.  Implementation and evaluation of SDPA 6.0 (Semidefinite Programming Algorithm 6.0) , 2003, Optim. Methods Softw..

[12]  Lieven Vandenberghe,et al.  Discrete Transforms, Semidefinite Programming, and Sum-of-Squares Representations of Nonnegative Polynomials , 2006, SIAM J. Optim..

[13]  Jos F. Sturm,et al.  A Matlab toolbox for optimization over symmetric cones , 1999 .

[14]  Ragnar Wallin,et al.  User's Guide to Kypd_Solver , 2003 .

[15]  B. Wahlberg,et al.  Modelling and Identification with Rational Orthogonal Basis Functions , 2000 .

[16]  Lieven Vandenberghe,et al.  Convex optimization problems involving finite autocorrelation sequences , 2002, Math. Program..

[17]  P. Parrilo.  Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization , 2000 .

[18]  A. Megretski,et al.  A new barrier function for IQC optimization problems , 2003, Proceedings of the 2003 American Control Conference, 2003..

[19]  Babak Hassibi,et al.  Indefinite-Quadratic Estimation And Control , 1987 .

[20]  Panos J. Antsaklis,et al.  Polynomial and rational matrix interpolation: theory and control applications , 1993 .

[21]  M. Rotea,et al.  LFTB: An Efficient Algorithm to Bound Linear Fractional Transformations , 2005 .

[22]  Vlad Ionescu,et al.  Continuous and discrete-time Riccati theory: A Popov-function approach , 1993 .

[23]  Johan Löfberg,et al.  YALMIP : a toolbox for modeling and optimization in MATLAB , 2004 .

[24]  C. Hol.  Structured controller synthesis for mechanical servo-systems: Algorithms, relaxations and optimality certificates , 2006 .

[25]  Xiong Zhang,et al.  Solving Large-Scale Sparse Semidefinite Programs for Combinatorial Optimization , 1999, SIAM J. Optim..

[26]  Kim-Chuan Toh,et al.  Solving semidefinite-quadratic-linear programs using SDPT3 , 2003, Math. Program..

[27]  L. Vandenberghe,et al.  Interior-Point Algorithms for Semidefinite Programming Problems Derived from the KYP Lemma , 2005 .

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

[29]  Johan Efberg,et al.  YALMIP : A toolbox for modeling and optimization in MATLAB , 2004 .

[30]  U.T. Jonsson,et al.  An algorithm for solving optimization problems involving special frequency dependent LMIs , 2000, Proceedings of the 2000 American Control Conference. ACC (IEEE Cat. No.00CH36334).

[31]  P. Parrilo.  On the numerical solution of LMIs derived from the KYP lemma , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[32]  A. Hansson,et al.  Polynomial complexity for a Nesterov-Todd potential-reduction method with inexact search directions , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).