Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones

SeDuMi is an add-on for MATLAB, that lets you solve optimization problems with linear, quadratic and semideeniteness constraints. It is possible to have complex valued data and variables in SeDuMi. Moreover, large scale optimization problems are solved eeciently, by exploiting sparsity. This paper describes how to work with this toolbox. SeDuMi stands for Self-Dual-Minimization: it implements the self-dual embedding technique for optimization over self-dual homogeneous cones. The self-dual embedding technique as proposed Ye, Todd and Mizuno 23], essentially makes it possible to solve certain optimization problems in a single phase, leading either to an optimal solution, or a certiicate of infeasibility. Optimization over self{dual homogeneous cones, or more concisely, optimization over symmetric cones, was rst studied by Nesterov and Todd 16], and is currently an active area of research. Semideenite programming is a special case of optimization over symmetric cones. The popular package SP by Vandenberghe and Boyd 21] is one of the rst software tools that was developed for semideenite programming. Some control theorists use SP indirectly via LMITOOL, by El Ghaoui, Nikoukhah and Delebecque 6], or MRCT, by Dussy and El Ghaoui 4], which are user-friendly front-ends for SP. A more recent and faster solver for semideenite programming is SDPA, by Fukisawa, Kojima and Nakata 8]. For optimization over symmetric cones, there are currently two software tools available, viz. SDPPack, by Alizadeh et al. 1], and SeDuMi. Both operate under the MATLAB environment, so that they can easily be used within speciic applications. SeDuMi has some features that are not available in SDPPack, namely it MATLAB is a registered trademark of The MathWorks, Inc. 1 allows the use of complex valued data, generates Farkas-dual solutions for infeasible problems, takes full advantage of sparsity, leading to signiicant speed beneets, has a theoretically proven O(p n log(1==)) worst-case iteration bound, can import linear programs in MPS format (via a link with LIPSOL 24]), and semideenite programs in SDPA 8] format. It is also possible to convert optimization problems from SDPPack 1] format into SeDuMi.