Doğrusal olmayan optimizasyon problemleri için taşınır algoritmik fonksiyonlar yöntemi

Bu çalışmada, sürekli gerçek sayılı fonksiyonların optimizasyon problemlerinin çözümüne yönelik olarak bir denklem önerilmiştir. Doğrudan analitik ya da sayısal çözümlere dönüşebildiği gibi, analitik veya sayısal yaklaşımla üretilebilecek yeni algoritma fonksiyonlarına da taşınabilen diferansiyel formdaki bu denklem, Taşınır Algoritmik Fonksiyonlar (TAF) Denklemi olarak isimlendirilmiştir. TAF denkleminin doğrusal olmayan sürekli fonksiyonlarla ifade edilen mühendislik problemlerine uygulanabilmesi mümkündür. Belirleyici (deterministik) olarak nitelendirilebilecek teknikler veren bu denklemle, olasılıksal (stochastic) yöntemlerde olduğu gibi istenirse türev bilgisi kullanmayan basit TAF yöntemleri tanımlanabilmektedir. Denklemin ve önerilen tekniklerin etkinliği uygulamalarla gösterilmiştir.

Transitive algorithmic functions method for nonlinear optimization problems

In this work, an equation is proposed to solve the general nonlinear continuous optimization problems. This equation can be used directly for obtaining analytical and numerical solutions. Since it can be transformed to some algorithmic functions through both analytically and numerically, this equation is named as the Transitive Algorithmic Functions (TAF) equation. The TAF equation can be applied to the optimization of general nonlinear continuous engineering problems. This equation presents alternative deterministic techniques and can define simple TAF algorithms which do not require derivative information as stochastic methods. Effectiveness of the equation and the proposed algorithms are demonstrated by their implementations.

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[1] Nocedal, J. ve Wright, S. J., “Numerical Optimization”, Springer, 2006.
[2] Dutta, J., “Optimization Theory - A Modern face of Applied Mathematics”, Directions, Indian Institute of Technology Kanpur, no. 3, vol. 6, 2004.
[3] Rao, S.S., “Engineering Optimization: Theory and Practice, Fourth Edition”, John Wiley and Sons, Inc., 2009.
[4] Yamashita, H., A Differential Equation Approach to Nonlinear Programming, Mathematical Programming, cilt 18, s.155-168, 1980.
[5] Absil, P.-A. ve Kurdyka, K., “On the stable equilibrium points of gradient systems”, Systems & Control Letters, cilt 55, 2006.
[6] Wang, S., Yang, X.Q. ve Teo, K. L., “A Unified Gradient Flow Approach to Constrained Nonlinear Optimization Problems”, Computational Optimization and Applications, cilt l. 25, no. 1-3, s.251-268, 2003.
[7] Masuda, K. ve Kurihara, K., “Global Optimization by Equilibrium-Point Search of Gradient-Based Dynamical System”, Electronics and Communications in Japan, cilt 91, no. 1, s.19–31, 2008.
[8] Bolte, J., “Continuous Gradient Projection Method in Hilbert Spaces”, Journal of Optimization Theory and Applications, cilt 119, no.2, s.235-259, 2003.
[9] Bian, W. ve Xue, X., “A Dynamical Approach to Constrained Nonsmooth Convex Minimization Problem Coupling with Penalty Function Method in Hilbert Space”, Numerical Functional Analysis and Optimization, cilt 31, no. 11, s. 1221-1253, 2010.
[10] Ozdemir, N. ve Evirgen F., “A Dynamic System Approach to Quadratic Programming Problems with Penalty Method”, Bulletin of the Malaysian Mathematical Sciences Society, cilt 33, no. 1, s.79-91, 2010.
[11] Xie, X., “L-BFGS and Delayed Dynamical Systems Approach for Unconstrained Optimization”, Postgraduate Research Symposium, Department of Computer Science, Hong Kong Baptist University, 2010.
[12] Alvarez, F, “On the Minimizing Property of a Second Order Dissipative System in Hilbert Spaces”, SIAM Journal on Control and Optimization, cilt 38, no.4; s.1102-1119, 2000.
[13] Alvarez, F., Attouch, H., Bolte, J. Ve Redont, P., “A Second-Order Gradient-Like Dissipative Dynamical System with Hessian-Driven Damping.: Application to Optimization and Mechanics”, Journal des Mathématiques Pures et Appliqués, cilt 81, no.8, s.747-779, 2002.
[14] Goudou, X. ve Munier, J., “The Gradient and Heavy Ball with Friction Dynamical Systems: the Quasiconvex Case”, Mathematical Programming, cilt 116, no.1-2, s.173-191, 2009.
[15] Jordan, D.W., and Smith, P., “Nonlinear Ordinary Differential Equations”, Oxford University Press, 2007.
[16] Hacıoğlu, A., ve Özkol, İ., “Vibrational Genetic Algorithm As a New Concept in Airfoil Design”, Aircraft Engineering and Aerospace Technology, cilt 3, no.3, s.228-236, 2002.
[17] Hacıoğlu, A., “Aerodinamik Dizayn ve Optimizasyonda Genetik Algoritma Kullanımı,” Uçak Mühendisliği Programı Doktora Tezi, İ.T.Ü. Fen Bilimleri Enstitüsü, İstanbul, 2003.