Summary: The problem of the best approximation with generalized restrictions is investigated. Under the assumption that \(l\) and \(u\) have finite intersection points, by introducing the concept of the sub-strong interior point condition and making use of the strong-CHIP from optimization theory, we study the relationship of the sub-strong interior point, the strong-CHIP and the characterization of the best approximation with generalized restrictions. Consequently, the characterization theorems of Kolmogorov’s type and “\(0\) belonging to the convex hull” type are obtained.