In order to produce precise enclosures from a multi-dimensional interval vector, we introduce a sharp interval sub-dividing condition for optimization algorithms. By utilizing interval inclusion properties, we also enhance the sampling of an upper bound for effective use in the interval quadratic method. This has resulted in an improvement in the algorithm for the unconstrained optimization problem by Hansen in 1992.

The knowledge of a good enclosure of the range of a function over small interval regions allows us to avoid convergence of optimization algorithms to a non-global point(s). We used interval slopes f[X,x] to check for monotonicity and integrated their derivative forms g[X,x], x X by quadratic and Newton methods to obtain narrow enclosures. In order to include boundary points in the search for the optimum point(s), we expanded the initial box by a small width on each dimension. These procedures resulted in an improvement in the algorithm proposed by Hansen.