\chapter{Conclusion}

For the most part, we consider this project to have been a success. Using very rigorous mathematical tools, we showed that the problem of consistency for a pair of interval-valued function and rectangular derivative approximation, defined on an $n$-dimensional domain, $n\ge 2$, reduces to whether a finite set of inequalities are simultaneously satisfied. In addition, we also proved that the minimal and maximal surfaces that are witnessing consistency can be derived by means of a linear programming algorithm, which provides a practical setting for implementation.
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By leveraging the Python programming language, we successfully developed a framework which implements the above linear test for consistency, given constraints for both function and derivative information within a partitioning of the $n$-dimensional unit hyper-rectangle. Moreover, the least and greatest surfaces are also determined whenever the provided input is found to be consistent. In order to provide a better insight into the problem, we implemented a simple graphical user interface which can depict the 3D piecewise linear surfaces whenever there is a consistent pair defined in the unit square $[0, 1]\times [0, 1]$.
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Finally, we spent a great deal of time developing an automatic testing framework that was crucial for assessing the correctness of our implementation. At the same time, we do not rule out the possibility of our implementation being completely error-free; however, we believe that the absence of any counterexamples in the sustained testing performed up to $6^{\rm{th}}$ dimension is convincing enough to guarantee a solid and reliable end-product.

\section{Future Work}

As an open-ended project, we suggest a list of possible ideas in which the present work can be further extended:

\begin{itemize}
	\item A faster method of input generation and translation of the linear programming problem into matrix form would be highly desirable. This may allow further test cases to be more quickly generated and verified.
	\item Extend the current framework to implement the algorithms that decide consistency in the case of a triangle/convex quadrilateral subject to convex derivative constraints.
	\item Also, examine if the latter algorithms in the convex setting can be extended to higher dimensions, similar to the rectangular case.
\end{itemize}
 