# Automatic Convexity Deduction for Efficient Function’s Range Bounding

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## Abstract

**:**

## 1. Introduction

- $\mathbb{R}$ — the set of real numbers;
- $\mathbb{Z}$ — the set of integers;
- $\mathbb{N}$ — the set of positive integers (natural numbers);
- $\mathbb{IR}$ — the set of all intervals in $\mathbb{R}$;
- $\mathbf{x}=[\underline{x},\overline{x}]$ — intervals are denoted with bold font;
- ${\mathcal{R}}_{f}\left([a,b]\right)=\{y\in \mathbb{R}:y=f\left(x\right)\phantom{\rule{0.277778em}{0ex}}\mathrm{for\; some}\phantom{\rule{0.277778em}{0ex}}x\in [a,b]\}$ — the range of function $f:\mathbb{R}\to \mathbb{R}$ over interval $[a,b]$;
- $\mathbf{f}$—an interval extension of a function $f:\mathbb{R}\to \mathbb{R}$, i.e., a mapping $\mathbf{f}:\mathbb{IR}\to \mathbb{IR}$ such that ${\mathcal{R}}_{f}\left([a,b]\right)$ $\subseteq \mathbf{f}\left(\right[a,b\left]\right)$ for any $[a,b]\in \mathbb{IR}$, notice, there may be many different interval extensions for a function $f\left(x\right)$;
- $f\left(x\right)\nearrow $—$f\left(x\right)$ is non-decreasing monotonic on $\mathbb{R}$ or an interval if additionally specified;
- $f\left(x\right)\searrow $—$f\left(x\right)$ is non-increasing monotonic on $\mathbb{R}$ or an interval if additionally specified.

## 2. Automatic Deduction of the Convexity and Concavity of a Function

#### 2.1. Deducing Monotonicity

**Proposition 1.**

- if $f\left(x\right)\nearrow $ on $[a,b]$ then $-f\left(x\right)\searrow $ on $[a,b]$;
- if $f\left(x\right)\searrow $ on $[a,b]$ then $-f\left(x\right)\nearrow $ on $[a,b]$;
- if $f\left(x\right)\nearrow $ and $g\left(x\right)\nearrow $ on $[a,b]$ then $f\left(x\right)+g\left(x\right)\nearrow $ on $[a,b]$;
- if $f\left(x\right)\nearrow $, $f\left(x\right)\ge 0$ and $g\left(x\right)\nearrow $, $g\left(x\right)\ge 0$ on $[a,b]$ then $f\left(x\right)\xb7g\left(x\right)\nearrow $ on $[a,b]$;
- if $f\left(x\right)\nearrow $ and $g\left(x\right)\nearrow $ on $[a,b]$ then $min\left(f\right(x),g(x\left)\right)\nearrow $ on $[a,b]$;
- if $f\left(x\right)\nearrow $ and $g\left(x\right)\nearrow $ on $[a,b]$ then $max\left(f\right(x),g(x\left)\right)\nearrow $ on $[a,b]$.

**Proposition 2.**

- If $h\left(x\right)\nearrow $ on $[a,b]$, $g\left(x\right)\nearrow $ on $[c,d]$ and ${\mathcal{R}}_{h}\left([a,b]\right)\subseteq [c,d]$ then $f\left(x\right)\nearrow $ on $[a,b]$.
- If $h\left(x\right)\searrow $ on $[a,b]$, $g\left(x\right)\nearrow $ on $[c,d]$ and ${\mathcal{R}}_{h}\left([a,b]\right)\subseteq [c,d]$ then $f\left(x\right)\searrow $ on $[a,b]$.
- If $h\left(x\right)\nearrow $ on $[a,b]$, $g\left(x\right)\searrow $ on $[c,d]$, ${\mathcal{R}}_{h}\left([a,b]\right)\subseteq [c,d]$ then $f\left(x\right)\searrow $ on $[a,b]$.
- If $h\left(x\right)\searrow $ on $[a,b]$, $g\left(x\right)\searrow $ on $[c,d]$, ${\mathcal{R}}_{h}\left([a,b]\right)\subseteq [c,d]$ then $f\left(x\right)\nearrow $ on $[a,b]$.

**Example 1.**

**Proposition 3.**

#### 2.2. Deducing Convexity

**Proposition 4.**

- $f\left(x\right)+g\left(x\right)$ is convex on $[a,b]$,
- $\alpha f\left(x\right)$ is convex on $[a,b]$ if $\alpha >0$,
- $max\left(f\right(x),g(x\left)\right)$ is convex on $[a,b]$.

**Proposition 5.**

**Proof.**

**Proposition 6.**

- g is convex and nondecreasing on $[c,d]$, h is convex on $[a,b]$, then f is convex on $[a,b]$,
- g is convex and nonincreasing on $[c,d]$, h is concave on $[a,b]$, then f is convex on $[a,b]$,
- g is concave and nondecreasing on $[c,d]$, h is concave on $[a,b]$, then f is concave on $[a,b]$,
- g is concave and nonincreasing on $[c,d]$, h is convex on $[a,b]$, then f is concave on $[a,b]$.

**Example 2.**

**Example 3.**

## 3. Application to Bounding the Function’s Range

**Proposition 7.**

**Proof.**

**Proposition 8.**

**Proof.**

**Proposition 9.**

**Proof.**

**Proposition 10.**

**Proof.**

## 4. Numerical Experiments

#### 4.1. Comparison with Interval Bounds

**Example 4.**

- $cos\left(x\right)$ is concave on $[0,1]$,
- $-cos\left(x\right)$ is convex on $[0,1]$ (by definition),
- x is concave on $[0,1]$,
- $-x$ is convex on $[0,1]$ (by definition),
- ${e}^{-x}$ is convex on $[0,1]$ (by Proposition 6),
- $-cos\left(x\right)+{e}^{-x}$ is convex on $[0,1]$ (by Proposition 4).

**Natural**—a bound computed by the natural interval expansion techniques,**Taylor**—a bound computed by the 1st order Taylor expansion,**Convex**—a bound computed according to Propositions 9 and 10.

#### 4.2. Impact on the Performance of Global Search

**Natural**—the natural interval expansion techniques,**Taylor**—the 1-st order Taylor expansion,**Convex**—the range is computed according to Propositions 3, 9 and 10.

**Natural**—pure natural interval expansion;**Natural + Convex**—the natural interval expansion combined with the proposed techniques;**Natural + Taylor**—the natural interval expansion combined with the first-order Taylor expansion;**Natural + Taylor + Convex**—the natural interval expansion combined with the first-order Taylor expansion and the proposed techniques.

## 5. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The overestimator (green) and the underestimator (red) of a convex function $f\left(x\right)$ on an interval $[a,b]$.

Type | Smooth | Non-Smooth |
---|---|---|

One variable | ${x}^{n}$, $\sqrt[n]{x}$, $1/x$, $ln\left(x\right)$, ${e}^{x}$ | $\left|x\right|$ |

$sin\left(x\right)$, $arcsin\left(x\right)$, $arctan\left(x\right)$ | ||

Two variables | $x+y$, $x\xb7y$ | $max(x,y)$, $max(x,y)$ |

Function | Increase | Decrease |
---|---|---|

$\left|x\right|$ | $[0,\infty )$ | $(-\infty ,0]$ |

${x}^{2n+1}$, $n\in \mathbb{N}$, ${e}^{x}$, | $(-\infty ,\infty )$ | — |

${x}^{2n}$, $n\in \mathbb{N}$ | $[0,\infty )$ | $(-\infty ,0]$ |

$\sqrt[n]{x}$ | $[0,\infty )$ | — |

$ln\left(x\right)$ | $(0,\infty )$ | — |

$1/x$ | — | $(-\infty ,0)\cup (0,\infty )$ |

$sin\left(x\right)$ | $[-\pi /2+2\pi k,\pi /2+2\pi k]$, $k\in \mathbb{Z}$ | $[\pi /2+2\pi k,3\pi /2+2\pi k]$, $k\in \mathbb{Z}$ |

$arcsin\left(x\right)$ | $[-1,1]$ | — |

$arctan\left(x\right)$ | $(-\infty ,\infty )$ | — |

Function | Convex | Concave |
---|---|---|

$\left|x\right|$, ${x}^{2n}$, $n\in \mathbb{N}$, ${e}^{x}$ | $(-\infty ,\infty )$ | — |

${x}^{2n+1}$, $n\in \mathbb{N}$, | $[0,\infty )$ | $(-\infty ,0]$ |

$\sqrt[n]{x}$ | — | $[0,\infty )$ |

$ln\left(x\right)$ | — | $(0,\infty )$ |

$1/x$ | $(0,\infty )$ | $(-\infty ,0)$ |

$sin\left(x\right)$ | $[-\pi +2\pi k,2\pi k]$, $k\in \mathbb{Z}$ | $[2\pi k,\pi +2\pi k]$, $k\in \mathbb{Z}$ |

$arcsin\left(x\right)$ | $[0,1]$ | $[-1,0]$ |

$arctan\left(x\right)$ | $(-\infty ,0]$ | $[0,\infty )$ |

**Table 4.**Comparison of natural interval expansion (Natural) Taylor expansion (Taylor) and bounds produced by the proposed techniques (Convex).

No | $\mathit{f}\left(\mathit{x}\right)$ | $[\mathit{a},\mathit{b}]$ | Natural | Taylor | Convex |
---|---|---|---|---|---|

1 | $-cos\left(x\right)+{e}^{-x}$ | $[0,1]$ | $[-0.632,0.46]$ | $[-0.77,0.23]$ | $[-0.438,0]$ |

2 | ${e}^{x}+{e}^{-x}$ | $[-0.5,0.5]$ | $[1.21,3.29]$ | $[1.48,2.52]$ | $[1.73,2.26]$ |

3 | $0.2{x}^{2}-sin\left(x\right)$ | $[0,\pi /2]$ | $[-1,0.49]$ | $[-1.37,0.2]$ | $[-0.92,0]$ |

4 | $2{\left(\right)}^{x}2$ | $[1,3]$ | $[1.65,98.02]$ | $[-260.66,279.44]$ | $[-0.92,90.02]$ |

5 | ${\left(\right)}^{x}2+{e}^{\left|x\right|}$ | $[-2,2]$ | $[9,79.39]$ | — | $[16.61,47.39]$ |

No | $\mathit{f}\left(\mathit{x}\right)$ | $[\mathit{a},\mathit{b}]$ | $\mathit{f}\left({\mathit{x}}_{*}\right)$ |
---|---|---|---|

1 | ${\left(\right)}^{x}2$ | $[-10,10]$ | 0 |

2 | $24{x}^{4}-142{x}^{3}+303{x}^{2}-276x+3$ | $[0,3]$ | 1 |

3 | ${x}^{4}-12{x}^{3}+47{x}^{2}-60x-20{e}^{-x}$ | $[-1,7]$ | $-32.781261$ |

4 | ${x}^{4}-10{x}^{3}+35{x}^{2}-50x+24$ | $[-10,20]$ | $-1$ |

5 | $-1.5{sin}^{2}\left(x\right)+sin\left(x\right)cos\left(x\right)+1.2$ | $[0.2,7]$ | $-0.451388$ |

6 | $-x+sin\left(\right)open="("\; close=")">3x$ | $[0.2,7]$ | $-5.815675$ |

7 | $x+sin\left(\right)open="("\; close=")">5x$ | $[0.2,7]$ | $-0.07759$ |

8 | $-sin\left(\right)open="("\; close=")">5x$ | $[0.2,7]$ | $-0.952897$ |

9 | $2cos\left(x\right)+cos\left(\right)open="("\; close=")">2x$ | $[0.2,7]$ | $3.5$ |

10 | $2{e}^{-x}sin\left(x\right)$ | $[0.2,7]$ | $-0.027864$ |

11 | $\left|x\right|+\left(\right)open="|"\; close="|">x-4$ | $[-8,8]$ | 8 |

12 | ${\left(\right)}^{x}2+{e}^{\left|x\right|}$ | $[-4,8]$ | 33 |

13 | $\left(\right)open="|"\; close="|">(x-1)/4$ | $[-10,10]$ | 1 |

14 | $\left(\right)open="("\; close=")">10\left(\right)open="|"\; close="|">sin\left(\right)open="("\; close=")">x+1+1$ | $[-10,10]$ | 1 |

No | Natural | Natural + Convex | Natural + Taylor | Natural + Taylor + Convex |
---|---|---|---|---|

1 | 35 | 15 | 29 | 15 |

2 | 135,043 | 199 | 267 | 81 |

3 | 98,995 | 107 | 269 | 79 |

4 | 72,953 | 151 | 311 | 91 |

5 | 443 | 39 | 83 | 39 |

6 | 187 | 19 | 47 | 19 |

7 | 183 | 39 | 69 | 39 |

8 | 189 | 49 | 91 | 49 |

9 | 857 | 31 | 75 | 31 |

10 | 51 | 19 | 27 | 19 |

11 | 55 | 5 | — | — |

12 | 579 | 23 | — | — |

13 | 35 | 27 | — | — |

14 | 125 | 125 | — | — |

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Posypkin, M.; Khamisov, O.
Automatic Convexity Deduction for Efficient Function’s Range Bounding. *Mathematics* **2021**, *9*, 134.
https://doi.org/10.3390/math9020134

**AMA Style**

Posypkin M, Khamisov O.
Automatic Convexity Deduction for Efficient Function’s Range Bounding. *Mathematics*. 2021; 9(2):134.
https://doi.org/10.3390/math9020134

**Chicago/Turabian Style**

Posypkin, Mikhail, and Oleg Khamisov.
2021. "Automatic Convexity Deduction for Efficient Function’s Range Bounding" *Mathematics* 9, no. 2: 134.
https://doi.org/10.3390/math9020134