Randomized algorithms to solve parameter-dependent linear matrix inequalities and their computational complexity

The randomized algorithm of G. Calafiore and B. Polyak (2000), which consists of random sampling and sub-gradient descent, is analyzed in the case where it is used to solve parameter-dependent linear matrix inequalities. This paper shows that the expected time to achieve a solution is infinite if this algorithm is used in its original form. However, it is also shown that the algorithm can be improved so that its expected achievement time becomes finite. An explicit upper bound of the expected achievement time is given in a special case. A numerical example is provided.

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