The solution equivalence to general models for the RIM quantifier problem (Q2337905)

Summary: In earlier papers, we investigated the relationship between the minimax disparity minimum variance regular increasing monotone (RIM) quantifier problems. We also proved the equivalence of their solutions to minimum variance and minimax disparity RIM quantifier problems. We investigated the relationship between the minimax ratio and maximum entropy RIM quantifier problems and proved the equivalence of their solutions to the maximum entropy and minimax ratio RIM quantifier problems. \textit{X. Liu} [Int. J. Approx. Reasoning 48, No. 2, 598--627 (2008; Zbl 1194.90124)] proposed a general RIM quantifier determination model and proved it analytically by using the optimal control technique. We also gave the equivalence of solutions to the minimax problem for the RIM quantifier. Recently, we proposed a modified model for the general minimax RIM quantifier problem and provided correct formulation of the result of Liu. Thus, we examine the general minimum model for the RIM quantifier problem when the generating functions are Lebesgue integrable under the more general assumption of the RIM quantifier operator. We also provide a solution equivalent relationship between the general maximum model and the general minimax model for RIM quantifier problems, which is the corrected and generalized version of the equivalence of solutions to the general maximum model and the general minimax model for RIM quantifier problems of Liu's result.