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PERFORMANCE ASSESSMENT OF STEEPEST DESCENT METHOD CONSIDERING GRADIENT BASED LINE SEARCH CONDITIONS IN GEOMETRY FITTING OF 2D MEASURED PROFILES

Year 2021, Volume: 8 Issue: 15, 515 - 524, 31.12.2021
https://doi.org/10.54365/adyumbd.997113

Abstract

This paper focuses on exploring effect of the step length, used in the line search, computation techniques on the performance of steepest descent (SD) method in the geometry fitting of the 2D measured profiles. To this end, the three step length computation techniques or line search conditions accommodating Weak Wolfe (WWC), Strong Wolfe (SWC) and the exact minimizer finder have been implemented during the fitting process. To test the line search conditions performances, the 2D primitive geometry test set consisting of five different geometries such as circle, square, triangle, ellipse and rectangle have been employed. The 2D profiles of those geometries have been extracted using coordinate measuring machine (CMM) with high precision. For performance assessments, the total number of function evaluations when the SD method-line search condition combination satisfies the required converge tolerance have been used. By means of those data, the performance profiles have been created to conduct reliable and efficient assessments on the line search conditions. The results have shown that the step length computation technique plays a crucial role for the SD method performance. Based on the performance profiles, it has been determined that the fastest line search condition is the WWC. Besides that, it has been revealed that the optimum technique is the exact minimizer finder for the geometry fitting process in the study.

Supporting Institution

Herhangi bir destekleyen kurum yoktur.

Thanks

The author acknowledges Design and Manufacturing Technologies Research Laboratory, Innovative Technologies Application and Research Center, Suleyman Demirel University where the experimental work in this study was performed.

References

  • Cauchy A. Methode generale pour la resolution des systemes d’equations simultanees. Comp. Rend. Sci. Paris 1847; 25(2): 536-538.
  • Bento G., da Cruz Neto J.X., Santos P. An inexact steepest descent method for multicriteria optimization on riemannian manifolds. Journal of Optimization Theory and Applications 2013; 159(1): 108-124.
  • Haug E., Arora J., Matsui K. A steepest-descent method for optimization of mechanical systems. Journal of Optimization Theory and Applications 1976; 19(3): 401-424.
  • Liu X., Reynolds A.C. A multiobjective steepest descent method with applications to optimal well control. Computational Geosciences 2016; 20(2): 355-374.
  • Zhu L.M., Ding H., Xiong Y.L. A steepest descent algorithm for circularity evaluation. Computer-Aided Design 2003; 35(3): 255-265.
  • Quiroz E.P., Quispe E., Oliveira P.R. Steepest descent method with a generalized armijo search for quasiconvex functions on riemannian manifolds. Journal of mathematical analysis and applications 2008; 341(1): 467-477.
  • Samir C., Absil, P.A., Srivastava, A., Klassen, E. A gradient-descent method for curve fitting on riemannian manifolds. Foundations of Computational Mathematics 2012; 12(1): 49-73.
  • George S., Sabari M. Convergence rate results for steepest descent type method for nonlinear ill-posed equations. Applied Mathematics and Computation 2017; 294: 169-179.
  • Anjidani M., Effati S. Steepest descent method for solving zero-one nonlinear programming problems. Applied Mathematics and Computation 2007; 193: 197-202.
  • Abbasbandy S., Jafarian A. Steepest descent method for solving fuzzy nonlinear equations. Applied Mathematics and Computation 2006; 174: 669-675.
  • [Khan K., Lobiyal D. Performance evaluation of different optimization techniques for coverage and connectivity control in backbone based wireless networks. Wireless Personal Communications 2017; 96(3): 4329-4345.
  • Rojas-Labanda S., Stolpe M. Benchmarking optimization solvers for structural topology optimization. Structural and Multidisciplinary Optimization 2015; 52(3): 527-547.
  • Tangherloni A., Spolaor S., Cazzaniga P., Besozzi D., Rundo L., Mauri G., Nobile M.S. Biochemical parameter estimation vs. benchmark functions: A comparative study of optimization performance and representation design. Applied Soft Computing 2019; 81:105494.
  • Villaverde A.F., Frohlich F., Weindl D., Hasenauer J., Banga J.R. Benchmarking optimization methods for parameter estimation in large kinetic models. Bioinformatics 2019; 35(5): 830-838.
  • Diachin L.F., Knupp P., Munson T., Shontz S. A comparison of two optimization methods for mesh quality improvement. Engineering with Computers 2006; 22(2): 61-74.
  • Arsenault R., Poulin A., Cote P., Brissette F. Comparison of stochastic optimization algorithms in hydrological model calibration. Journal of Hydrologic Engineering 2014; 19(7): 1374-1384.
  • https://www.desmos.com (Access date:16.05.2021)
  • Jia P. Fitting a parametric model to a cloud of points via optimization methods. Ph.D. thesis. New York: Syracuse University; 2017.
  • Nocedal J., Wright S.J. Numerical optimization. 2nd ed. New York: Springer Science & Business Media; 2006
  • Dolan E.D., More J.J. Benchmarking optimization software with performance profiles. Mathematical programming 2002; 91(2): 201-213.

ÖLÇÜLEN 2B PROFİLLERE GEOMETRİ UYDURULMASINDA GRADYAN TEMELLİ DOĞRU BOYUNCA ARAMA ŞARTLARI DİKKATE ALINARAK EN DİK İNİŞ YÖNTEMİNİN PERFORMANS DEĞERLENDİRMESİ

Year 2021, Volume: 8 Issue: 15, 515 - 524, 31.12.2021
https://doi.org/10.54365/adyumbd.997113

Abstract

Bu çalışmada ölçülen 2B profillere geometri uydurulmasında doğru boyunca aramada kullanılan adım uzunluğu hesaplama yöntemlerinin en dik iniş yönteminin performansına etkisinin ortaya çıkarılmasına odaklanılmaktadır. Bu amaçla, zayıf Wolfe, güçlü Wolfe ve tam olarak minimize eden adım uzunluğunu bulan olmak üzere üç adım uzunluğu hesaplama yöntemi ya da doğru boyunca arama şartları geometri uydurma sürecinde kullanılmıştır. Doğru boyunca arama şartlarının performansını test etmek amacıyla, daire, kare, üçgen, elips ve dikdörtgen geometrilerini içeren bir 2B temel geometri seti kullanılmıştır. Bu geometrilerin profilleri yüksek hassasiyet ile koordinat ölçme cihazı ile elde edilmiştir. Performans değerlendirmeleri için ilgili en dik iniş yöntemi-doğru boyunca arama şartı kombinasyonu istenen tolerans değerini sağladığında ortaya çıkan toplam fonksiyon kullanım sayısı kullanılmıştır. Doğru boyunca arama şartlarının güvenilir ve verimli bir şekilde performans değerlendirilmesini yapmak için bu veriler vasıtasıyla performans profilleri oluşturulmuştur. Sonuçlar adım uzunluğu hesaplama tekniklerinin en dik iniş yöntemi performansında önemli bir rol oynadığını göstermektedir. Performans profillerine dayanarak, en hızlı doğru boyunca arama şartı zayıf Wolfe olarak saptanmıştır. Bunun yanı sıra, çalışmada geometri uydurma süreci için optimum yöntemin tam olarak minimize eden adım uzunluğunu bulan yöntem olduğu ortaya çıkarılmıştır.

References

  • Cauchy A. Methode generale pour la resolution des systemes d’equations simultanees. Comp. Rend. Sci. Paris 1847; 25(2): 536-538.
  • Bento G., da Cruz Neto J.X., Santos P. An inexact steepest descent method for multicriteria optimization on riemannian manifolds. Journal of Optimization Theory and Applications 2013; 159(1): 108-124.
  • Haug E., Arora J., Matsui K. A steepest-descent method for optimization of mechanical systems. Journal of Optimization Theory and Applications 1976; 19(3): 401-424.
  • Liu X., Reynolds A.C. A multiobjective steepest descent method with applications to optimal well control. Computational Geosciences 2016; 20(2): 355-374.
  • Zhu L.M., Ding H., Xiong Y.L. A steepest descent algorithm for circularity evaluation. Computer-Aided Design 2003; 35(3): 255-265.
  • Quiroz E.P., Quispe E., Oliveira P.R. Steepest descent method with a generalized armijo search for quasiconvex functions on riemannian manifolds. Journal of mathematical analysis and applications 2008; 341(1): 467-477.
  • Samir C., Absil, P.A., Srivastava, A., Klassen, E. A gradient-descent method for curve fitting on riemannian manifolds. Foundations of Computational Mathematics 2012; 12(1): 49-73.
  • George S., Sabari M. Convergence rate results for steepest descent type method for nonlinear ill-posed equations. Applied Mathematics and Computation 2017; 294: 169-179.
  • Anjidani M., Effati S. Steepest descent method for solving zero-one nonlinear programming problems. Applied Mathematics and Computation 2007; 193: 197-202.
  • Abbasbandy S., Jafarian A. Steepest descent method for solving fuzzy nonlinear equations. Applied Mathematics and Computation 2006; 174: 669-675.
  • [Khan K., Lobiyal D. Performance evaluation of different optimization techniques for coverage and connectivity control in backbone based wireless networks. Wireless Personal Communications 2017; 96(3): 4329-4345.
  • Rojas-Labanda S., Stolpe M. Benchmarking optimization solvers for structural topology optimization. Structural and Multidisciplinary Optimization 2015; 52(3): 527-547.
  • Tangherloni A., Spolaor S., Cazzaniga P., Besozzi D., Rundo L., Mauri G., Nobile M.S. Biochemical parameter estimation vs. benchmark functions: A comparative study of optimization performance and representation design. Applied Soft Computing 2019; 81:105494.
  • Villaverde A.F., Frohlich F., Weindl D., Hasenauer J., Banga J.R. Benchmarking optimization methods for parameter estimation in large kinetic models. Bioinformatics 2019; 35(5): 830-838.
  • Diachin L.F., Knupp P., Munson T., Shontz S. A comparison of two optimization methods for mesh quality improvement. Engineering with Computers 2006; 22(2): 61-74.
  • Arsenault R., Poulin A., Cote P., Brissette F. Comparison of stochastic optimization algorithms in hydrological model calibration. Journal of Hydrologic Engineering 2014; 19(7): 1374-1384.
  • https://www.desmos.com (Access date:16.05.2021)
  • Jia P. Fitting a parametric model to a cloud of points via optimization methods. Ph.D. thesis. New York: Syracuse University; 2017.
  • Nocedal J., Wright S.J. Numerical optimization. 2nd ed. New York: Springer Science & Business Media; 2006
  • Dolan E.D., More J.J. Benchmarking optimization software with performance profiles. Mathematical programming 2002; 91(2): 201-213.
There are 20 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Makaleler
Authors

Kadir Kıran 0000-0002-6109-435X

Publication Date December 31, 2021
Submission Date September 17, 2021
Published in Issue Year 2021 Volume: 8 Issue: 15

Cite

APA Kıran, K. (2021). PERFORMANCE ASSESSMENT OF STEEPEST DESCENT METHOD CONSIDERING GRADIENT BASED LINE SEARCH CONDITIONS IN GEOMETRY FITTING OF 2D MEASURED PROFILES. Adıyaman Üniversitesi Mühendislik Bilimleri Dergisi, 8(15), 515-524. https://doi.org/10.54365/adyumbd.997113
AMA Kıran K. PERFORMANCE ASSESSMENT OF STEEPEST DESCENT METHOD CONSIDERING GRADIENT BASED LINE SEARCH CONDITIONS IN GEOMETRY FITTING OF 2D MEASURED PROFILES. Adıyaman Üniversitesi Mühendislik Bilimleri Dergisi. December 2021;8(15):515-524. doi:10.54365/adyumbd.997113
Chicago Kıran, Kadir. “PERFORMANCE ASSESSMENT OF STEEPEST DESCENT METHOD CONSIDERING GRADIENT BASED LINE SEARCH CONDITIONS IN GEOMETRY FITTING OF 2D MEASURED PROFILES”. Adıyaman Üniversitesi Mühendislik Bilimleri Dergisi 8, no. 15 (December 2021): 515-24. https://doi.org/10.54365/adyumbd.997113.
EndNote Kıran K (December 1, 2021) PERFORMANCE ASSESSMENT OF STEEPEST DESCENT METHOD CONSIDERING GRADIENT BASED LINE SEARCH CONDITIONS IN GEOMETRY FITTING OF 2D MEASURED PROFILES. Adıyaman Üniversitesi Mühendislik Bilimleri Dergisi 8 15 515–524.
IEEE K. Kıran, “PERFORMANCE ASSESSMENT OF STEEPEST DESCENT METHOD CONSIDERING GRADIENT BASED LINE SEARCH CONDITIONS IN GEOMETRY FITTING OF 2D MEASURED PROFILES”, Adıyaman Üniversitesi Mühendislik Bilimleri Dergisi, vol. 8, no. 15, pp. 515–524, 2021, doi: 10.54365/adyumbd.997113.
ISNAD Kıran, Kadir. “PERFORMANCE ASSESSMENT OF STEEPEST DESCENT METHOD CONSIDERING GRADIENT BASED LINE SEARCH CONDITIONS IN GEOMETRY FITTING OF 2D MEASURED PROFILES”. Adıyaman Üniversitesi Mühendislik Bilimleri Dergisi 8/15 (December 2021), 515-524. https://doi.org/10.54365/adyumbd.997113.
JAMA Kıran K. PERFORMANCE ASSESSMENT OF STEEPEST DESCENT METHOD CONSIDERING GRADIENT BASED LINE SEARCH CONDITIONS IN GEOMETRY FITTING OF 2D MEASURED PROFILES. Adıyaman Üniversitesi Mühendislik Bilimleri Dergisi. 2021;8:515–524.
MLA Kıran, Kadir. “PERFORMANCE ASSESSMENT OF STEEPEST DESCENT METHOD CONSIDERING GRADIENT BASED LINE SEARCH CONDITIONS IN GEOMETRY FITTING OF 2D MEASURED PROFILES”. Adıyaman Üniversitesi Mühendislik Bilimleri Dergisi, vol. 8, no. 15, 2021, pp. 515-24, doi:10.54365/adyumbd.997113.
Vancouver Kıran K. PERFORMANCE ASSESSMENT OF STEEPEST DESCENT METHOD CONSIDERING GRADIENT BASED LINE SEARCH CONDITIONS IN GEOMETRY FITTING OF 2D MEASURED PROFILES. Adıyaman Üniversitesi Mühendislik Bilimleri Dergisi. 2021;8(15):515-24.