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We (see Adams et al., ibid., p.314, 1994) presented in an earlier article a new quadratic programming algorithm that can design constrained least-squares FIR digital filters. In this article we present a new quadratic programming algorithm for designing constrained minimax FIR filters. The new algorithm is better than the Parks-McClellan (1973) algorithm because it can design a much wider variety of filters. In particular, it can do minimax optimization subject to arbitrary equality and inequality constraints.