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For solving linear variational inequalities(LVIs) and quadratic optimization problems(QOPs), a new delayed projection neural network is proposed in this paper. And some sufficient conditions ensuring exponential stability are obtained via constructing appropriate Lyapunov functionals. As a special case, a matrix constraint is considered too. In this case, by dividing the network state variables into subgroups according to the character of the activation functions, some more compact sufficient conditions ensuring exponential stability are obtained, and these conditions are only relate to some blocks of the interconnection matrix. One numerical example will be presented, to show the effectiveness of the main results.