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About using LP programing to solve LMI with singular matrix http://forum.alglib.net/viewtopic.php?f=2&t=4467 
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Author:  Quentin987 [ Tue Jun 28, 2022 6:18 pm ] 
Post subject:  About using LP programing to solve LMI with singular matrix 
Hello I have a question concerning the use of LP programing. I would like to solve the LMI problem of the form C.M+M^t.C<0 with the constraint C>0. M is a square matrix with numerical values and C is a diagonal matrix with c_i>0 to be determined. The issue is that the matrix M is such that det M =0. (these conditions arise from the Liapunov function to check whether the dynamical system is stable). So in that case, does the algorithm works? also can the problem be implemented in its current formulation or is it required to reformulate it following the form Ax<0 with x>0? It is the first time a try to solve this type of constraint. the matrix M is too big to be handled manually (M = 63*63), so I am looking for the algorithmic method. thank you for any comments. regards Quentin 
Author:  Sergey.Bochkanov [ Fri Jul 01, 2022 6:33 pm ] 
Post subject:  Re: About using LP programing to solve LMI with singular mat 
Hi! Assuming that "C.M+M^t.C<0" means componentwise inequalities, the answer is: yes, it is possible to solve with ALGLIB interior point method, and the fact that M is degenerate should not result in algorithm failure. ALGLIB IPM employs regularization anyway. However, you have to reformulate your matrix inequality to Ax<0  presently ALGLIB supports only simple inequalities. 
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