**5. Discussion**

The standard way to ensure the convexity is to bound the range of the function's second derivative. However, this approach is only applicable to smooth functions. We defined a set of rules that can efficiently handle nonsmooth functions. The algebraic representation for the function, however, should be available.

It is worth noting that the proposed approach can be efficiently coded in modern programming languages supporting the operator's overloading techniques. To run the experiments presented in Tables 4 and 6 we have implemented our approach in Python. The elementary functions and operators were overloaded to support a particular data type that carries monotonicity and convexity information and the range of the function. The overloaded methods work according to the rules described in Section 2 and interval arithmetic.

As we have shown above, evaluating convexity can improve interval bounds on the function's range and accelerate the global optimization algorithms. Moreover, the over- and underestimators defined by the Propositions 7 and 8 enable efficient reduction techniques. The reduction techniques are widely used to accelerate the search for a global minimum of a function or a root of an equation.

We believe that the proposed approach has great potential as it can be extended to various generalized notions of convexity, e.g., quasiconvexity [46]. Quasiconvex functions possess the unimodality property, and thus recognizing the quasiconvexity (quasiconcavity) can tremendously enhance global optimization algorithms.

**Author Contributions:** Investigation: both authors. All authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was supported by the Ministry of Science and Higher Education of the Russian Federation, project No 075-15-2020-799.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


46. Hadjisavvas, N.; Komlósi, S.; Schaible, S.S. *Handbook of Generalized Convexity and Generalized Monotonicity*; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2006; Volume 76.
