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An ellipsoid-based randomized algorithm of Kanev et al. is extended for the use of parameter-dependent Lyapunov functions. The proposed algorithm is considered to be useful for a less conservative design of a robust state-feedback controller against nonlinear parametric uncertainty. Indeed, it enables us to avoid polytopic overbounding of uncertainty and employment of parameter-independent Lyapunov functions. After a bounded number of iterations, the proposed algorithm gives with high confidence a probabilistic solution that satisfies a provided inequality for a high-percentage of parameters. This algorithm can be used also for finding an optimal solution in an approximated sense. Convergence to a non-strict deterministic solution is considered and, especially, the expected number of iterations necessary to achieve a non-strict deterministic solution is provided infinite under some assumptions. A numerical example is provided.