# A Generator of Nonregular Semidefinite Programming Problems

## Abstract

Regularity is an important property of optimization problems. Various notions of regularity are known from the literature, being defined for different classes of problems. Usually, optimization methods are based on the optimality conditions, that in turn, often suppose that the problem is regular. Absence of regularity leads to theoretical and numerical difficulties, and solvers may fail to provide a trustworthy result. Therefore, it is very important to verify if a given problem is regular in terms of certain regularity conditions and in the case of nonregularity, to apply specific methods. On the other hand, in order to test new stopping criteria and the computational behaviour of new methods, it is important to have an access to sets of reasonably-sized nonregular test problems. The paper presents a generator that constructs nonregular Semidefinite Programming (SDP) instances with prescribed irregularity degrees and a database of nonregular test problems created using this generator. Numerical experiments using popular SDP solvers on the problems of the database are carried out and permit to conclude that the most popular SDP solvers are not efficient when applied to nonregular problems.

## Keywords

Semidefinite programming Regularity Constraint qualification Good behaviour Generator of nonregular sdp problems## Notes

### Acknowledgements

The authors would like to thank the anonymous referees for their suggestions and valuable comments that have helped to improve the paper. This work was supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (FCT - Fundação para a Ciência e a Tecnologia), within project UID/MAT/04106/2013.

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