This paper proposes a generalized set-theoretic interval observer (GSIO) for robust state estimation of discrete linear time-invariant (LTI) systems with bounded uncertainties. Including the previous set-theoretic interval observer (SIO) as a special case, the proposed GSIO only requires to transform the system matrix of the first subsystem to be nonnegative. Based on this idea, a nonsmooth H∞ optimization method is used to design a nonnegative subsystem matrix for the first subsystem based on an element-wise nonnegativity transformation. The advantage of this method is that it can design a nonnegative subsystem matrix by transforming each of its elements to be nonnegative, which does not impact any structural constraints on matrix elements. This implies that more extra design degrees of freedom could be obtained from the coordinate transformation and observer gain matrices, which can optimize state estimation performance of GSIO (e.g., suppress the effects of disturbances and noises). In this way, we could not only achieve an expected balance between robust state estimation conservatism and computational complexity but also better robust state estimation performance. Besides, under the framework of the proposed GSIO, this paper further proves its existence, gives its stability condition and compares it with the set-valued observer (SVO) and the interval observer (IO) in state estimation conservatism, which together forms a complete theoretic system for the proposed GSIO. At the end of this paper, numerical examples are used to illustrate the effectiveness of the GSIO.