User's Guide for SEDUMI INTERFACE1.04

This report describes a user-friendly M ATLAB package for defining Linear Matrix Constraints (LMCs). It acts as an interface for the Self-Dual-Minimisation package (S EDUMI) developed by Jos F. Sturm. The functionalities of S EDUMI INTERFACE are the following: Declare an LMC problem. Five MATLAB functions allow to define completely an LMC problem which can be characterised by scalar and matrix variables, linear matrix equality (LME) constraints, linear matrix inequality (LMI) constraints and a linear objective: – Initialise the LMC problem: sdmpb. – Declare the matrix variables: sdmvar. – Declare the block partitioned equality constraints: sdmlme and sdmequ. – Declare the block partitioned inequality constraints: sdmlmi and sdminequ. – Declare the linear objective: sdmobj. Solve an LMC problem. A unique function, sdmsol, calls the S EDUMI solver. Options allow to tune the solver parameters. Modify an LMC problem. At any moment it is possible to append an LMC problem by adding variables, inequalities or linear terms to the objective. Moreover, the sdmset function allows to freeze matrix variables to specified values. Analyse the solution issued from the solver. For all (feasible or not) problems, the solver outputs the last computed iterate (sdmget). SEDUMI INTERFACEallows to analyse this result in a convivial display. The solution is displayed directly in matrix format and indicators show which constraints are satisfied.

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

[2]  Brian Borchers.  CSDP 2.3 user's guide , 1999 .

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

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

[5]  R. Braatz,et al.  A tutorial on linear and bilinear matrix inequalities , 2000 .

[6]  Dimitri Peaucelle,et al.  SEDUMI INTERFACE 1.02: a tool for solving LMI problems with SEDUMI , 2002, Proceedings. IEEE International Symposium on Computer Aided Control System Design.

[7]  S. J. Benson,et al.  DSDP3: dual scaling algorithm for general positive semidefinite programming. , 2001 .

[8]  Laurent El Ghaoui,et al.  Advances in linear matrix inequality methods in control: advances in design and control , 1999 .

[9]  L. Ghaoui,et al.  A cone complementarity linearization algorithm for static output-feedback and related problems , 1997, IEEE Trans. Autom. Control..

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

[11]  Friedemann Leibfritz,et al.  An LMI-Based Algorithm for Designing Suboptimal Static H2/Hinfinity Output Feedback Controllers , 2000, SIAM J. Control. Optim..

[12]  Stephen P. Boyd,et al.  Linear Matrix Inequalities in Systems and Control Theory , 1994 .

[13]  Stephen P. Boyd,et al.  Design and implementation of a parser/solver for SDPs with matrix structure , 1996, Proceedings of Joint Conference on Control Applications Intelligent Control and Computer Aided Control System Design.

[14]  Hans D. Mittelmann,et al.  An independent benchmarking of SDP and SOCP solvers , 2003, Math. Program..