This paper is devoted to a generalization of the non-standard Kalman Filter (KF) introduced in [4]. We deal with some restrictions of the technical resources in the context of a state estimation problem and study a constrained convex program. Moreover, we replace two main concepts of the conventional KF, namely, the fundamental Normality Hypothesis (NH) and the unconstrained optimization approach. The minimax methodology we propose make it possible to develop an effective quasi-explicit solution method for the practically motivated generalization of the Kalman-type filter. We present a rigorous formal analysis of the obtained algorithm. The resulting non-linear filter possesses a strong optimality properties.