# On One Problem of the Nonlinear Convex Optimization

## Abstract

**:**

## 1. Formulation of the Problem

- -
- They are close to being the broadest class of problems we know how to solve efficiently.
- -
- They enjoy nice geometric properties (e.g., local minima are global).
- -
- There are excellent softwares that readily solve a large class of convex problems.
- -
- Numerous important problems in a variety of application domains are convex.

#### A Brief History of Convex Optimization

**Theorem**

**1.**

## 2. Analytical and Numerical Solution of the Convex Optimization Problem

**Theorem**

**2.**

**Proof.**

**Remark**

**1.**

**Remark**

**2.**

**Listing 1.**MATLAB code used for calculating ${l}_{\mathrm{maxLength}}\left(\beta \right)$ for ${d}_{1}=1$, ${d}_{2}=2$ and $\beta =\frac{\pi}{2}$.

syms alpha beta d1 d2 |

d1 = 1; % width of the first channel |

d2 = 2; % width of the second channel |

beta = pi/2; % an angle between the navigable channels |

l = (d1/sin(alpha))+(d2/sin(alpha+beta)); % the cost function from (1) |

la = diff(l,alpha); |

eqn = la == 0; |

num = vpasolve(eqn,alpha,[0 pi-beta]); % numerical solver |

solution_max_length = simplify(subs(l,num),’Steps’,20) % output |

## 3. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Ahmadi, A.A. Princeton University. 2015. Available online: https://www.princeton.edu/~aaa/Public/Teaching/ORF523/S16/ORF523_S16_Lec4_gh.pdf (accessed on 10 August 2022).
- Stigler, S.M. Gauss and the Invention of Least Squares. Ann. Stat.
**1981**, 9, 465–474. [Google Scholar] [CrossRef] - Ghaoui, L.E. University of Berkeley. 2013. Available online: https://people.eecs.berkeley.edu/~elghaoui/Teaching/EE227BT/LectureNotes_EE227BT.pdf (accessed on 10 August 2022).
- Dantzig, G. Origins of the simplex method. In A History of Scientific Computing; Nash, S.G., Ed.; Association for Computing Machinery: New York, NY, USA, 1987; pp. 141–151. [Google Scholar]
- Dantzig, G. Linear Programming and Extensions; Princeton University Press: Princeton, NJ, USA, 1963. [Google Scholar]
- Fiacco, A.; McCormick, G. Nonlinear programming: Sequential Unconstrained Minimization Techniques; John Wiley and Sons: New York, NY, USA, 1968. [Google Scholar]
- Dikin, I. Iterative solution of problems of linear and quadratic programming. Sov. Math. Dokl.
**1967**, 174, 674–675. [Google Scholar] - Dikin, I. On the speed of an iterative process. Upr. Sist.
**1974**, 12, 54–60. [Google Scholar] - Yudin, D.; Nemirovskii, A. Informational complexity and efficient methods for the solution of convex extremal problems. Matekon
**1976**, 13, 22–45. [Google Scholar] - Khachiyan, L. A polynomial time algorithm in linear programming. Sov. Math. Dokl.
**1979**, 20, 191–194. [Google Scholar] - Shor, N. On the Structure of Algorithms for the Numerical Solution of Optimal Planning and Design Problems. Ph.D. Thesis, Cybernetic Institute, Academy of Sciences of the Ukrainian SSR, Kiev, Ukraine, 1964. [Google Scholar]
- Shor, N. Cut-off method with space extension in convex programming problems. Cybernetics
**1977**, 13, 94–96. [Google Scholar] [CrossRef] - Rodomanov, A.; Nesterov, Y. Subgradient ellipsoid method for nonsmooth convex problems. Math. Program.
**2022**, 1–37. [Google Scholar] [CrossRef] - Karmarkar, N. A new polynomial time algorithm for linear programming. Combinatorica
**1984**, 4, 373–395. [Google Scholar] [CrossRef] - Adler, I.; Karmarkar, N.; Resende, M.; Veiga, G. An implementation of Karmarkar’s algorithm for linear programming. Math. Program.
**1989**, 44, 297–335. [Google Scholar] [CrossRef] - Klee, V.; Minty, G. How Good Is the Simplex Algorithm? Inequalities—III; Shisha, O., Ed.; Academic Press: New York, NY, USA, 1972; pp. 159–175. [Google Scholar]
- Nesterov, Y.; Nemirovski, A.S. Interior Point Polynomial Time Methods in Convex Programming; SIAM: Philadelphia, PA, USA, 1994. [Google Scholar]
- Nemirovski, A.S.; Todd, M.J. Interior-point methods for optimization. Acta Numer.
**2008**, 17, 191–234. [Google Scholar] [CrossRef] - Nimana, N. A Fixed-Point Subgradient Splitting Method for Solving Constrained Convex Optimization Problems. Symmetry
**2020**, 12, 377. [Google Scholar] [CrossRef] - Han, D.; Liu, T.; Qi, Y. Optimization of Mixed Energy Supply of IoT Network Based on Matching Game and Convex Optimization. Sensors
**2020**, 20, 5458. [Google Scholar] [CrossRef] [PubMed] - Popescu, C.; Grama, L.; Rusu, C. A Highly Scalable Method for Extractive Text Summarization Using Convex Optimization. Symmetry
**2021**, 13, 1824. [Google Scholar] [CrossRef] - Jiao, Q.; Liu, M.; Li, P.; Dong, L.; Hui, M.; Kong, L.; Zhao, Y. Underwater Image Restoration via Non-Convex Non-Smooth Variation and Thermal Exchange Optimization. J. Mar. Sci. Eng.
**2021**, 9, 570. [Google Scholar] [CrossRef] - Alfassi, Y.; Keren, D.; Reznick, B. The Non-Tightness of a Convex Relaxation to Rotation Recovery. Sensors
**2021**, 21, 7358. [Google Scholar] [CrossRef] - Boyd, S.; Vandenberghe, L. Convex Optimization; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]
- Bartsch, H.-J.; Sachs, M. Taschenbuch Mathematischer Formeln für Ingenieure und Naturwissenschaftler, 24th ed.; Neu Bearbeitete Auflage; Carl Hanser Verlag GmbH & Co., KG.: Munich, Germany, 2018. [Google Scholar]
- Ebrahimnejad, A.; Verdegay, J.L. An efficient computational approach for solving type-2 intuitionistic fuzzy numbers based Transportation Problems. Int. J. Comput. Intell. Syst.
**2016**, 9, 1154–1173. [Google Scholar] [CrossRef] - Ebrahimnejad, A.; Nasseri, S.H. Linear programmes with trapezoidal fuzzy numbers: A duality approach. Int. J. Oper. Res.
**2012**, 13, 67–89. [Google Scholar] [CrossRef]

**Figure 2.**The objective function ${l}_{\beta}\left(\alpha \right)$ and its second derivative for $\beta =\frac{\pi}{2}$, ${d}_{1}=1$ and ${d}_{2}=1$ on the interval $[0,\pi -\beta ]$.

**Figure 3.**The minimum values of ${d}_{1}$ and ${d}_{2}$ for a 5-unit-long crossbar (d) to pass through the channel (with $\beta =\frac{\pi}{2}$).

**Table 1.**${l}_{\mathrm{maxLength}}\left(\beta \right)$ for the different values of the parameters by employing the code from Listing 1.

$\mathit{\beta}=\frac{\mathit{\pi}}{100}$ | $\mathit{\beta}=\frac{\mathit{\pi}}{6}$ | $\mathit{\beta}=\frac{\mathit{\pi}}{4}$ | $\mathit{\beta}=\frac{\mathit{\pi}}{2}$ | $\mathit{\beta}=\frac{2\mathit{\pi}}{3}$ | $\mathit{\beta}=\frac{3\mathit{\pi}}{4}$ | $\mathit{\beta}=\frac{99\ast \mathit{\pi}}{100}$ | |
---|---|---|---|---|---|---|---|

${d}_{1}=1$, ${d}_{2}=1$ | 2.00 | 2.07 | 2.16 | 2.82 | 4.00 | 5.22 | 127.32 |

${d}_{1}=1$, ${d}_{2}=3$ | 4.00 | 4.10 | 4.25 | 5.40 | 7.54 | 9.80 | 237.60 |

${d}_{1}=1$, ${d}_{2}=5$ | 6.00 | 6.12 | 6.29 | 7.77 | 10.69 | 13.84 | 333.35 |

${d}_{1}=2$, ${d}_{2}=7$ | 9.00 | 9.22 | 9.54 | 12.01 | 16.70 | 21.69 | 524.70 |

${d}_{1}=2$, ${d}_{2}=9$ | 11.00 | 11.23 | 11.58 | 14.38 | 19.84 | 25.71 | 620.27 |

${d}_{1}=2$, ${d}_{2}=11$ | 13.00 | 13.24 | 13.60 | 16.70 | 22.90 | 29.61 | 712.44 |

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**MDPI and ACS Style**

Vrabel, R.
On One Problem of the Nonlinear Convex Optimization. *AppliedMath* **2022**, *2*, 512-517.
https://doi.org/10.3390/appliedmath2040030

**AMA Style**

Vrabel R.
On One Problem of the Nonlinear Convex Optimization. *AppliedMath*. 2022; 2(4):512-517.
https://doi.org/10.3390/appliedmath2040030

**Chicago/Turabian Style**

Vrabel, Robert.
2022. "On One Problem of the Nonlinear Convex Optimization" *AppliedMath* 2, no. 4: 512-517.
https://doi.org/10.3390/appliedmath2040030