Derivative free optimization methods: Application in stirrer configuration and data clustering

Abstract

türevi doğrudan elde edilememektedir. Bu uygulama ile ilgili nümerik sonuçlar yorumlanarak sunulmaktadır. Bunlara ek olarak, çeşitli DFO araştırmalarının analizi ve bunların bir değerlendirmesi yapılarak bu konuyla ilgili çalışmalara katkıda bulunulmuştur. Ayrıca, türevsiz metotların hesaplamalarda başarılı sonuçlar verdiğini göstermek amacıyla pürüzlü (non-smooth) optimizasyon teknikleri ile birlikte kullanılan, türevsiz bir sınıflandırma algoritması incelenmektedir. Bu amaçla, bu algoritma günlük yaşamda ve tıpta sık sık karşılaşılan çeşitli veri kümelerine uygulanmaktadır. Bu metotları teorik olarak karşılaştırıp, türevsiz optimizasyon alanının şu anki genel görünümü ve özeti verilmektedir. Anahtar Kelimeler: türevsiz optimizasyon, trust-region metodu, well-poisedness, doğrusal olmayan optimizasyon, karıştırıcı konfigürasyonları, support vector machines, pürüzlü optimizasyon, ters problemler, istatiksel öğrenme, veri, sınıflandırma. vu

Recent developments show that derivative free methods are highly demanded by researches for solving optimization problems in various practical contexts. Although well-known optimization methods that employ derivative information can be very efficient, a derivative free method will be more efficient in cases where the objective function is nondifferentiable, the derivative information is not available or is not reliable. Derivative Free Optimization (DFO) is devel oped for solving small dimensional problems (less than 100 variables) in which the computation of an objective function is relatively expensive and the deriv atives of the objective function are not available. Problems of this nature more and more arise in modern physical, chemical and econometric measurements and in engineering applications, where computer simulation is employed for the evaluation of the objective functions. In this thesis, we give an example of the implementation of DFO in an approach for optimizing stirrer configurations, including a parametrized grid generator, a flow solver, and DFO. A derivative free method, i.e., DFO is preferred be cause the gradient of the objective function with respect to the stirrer's designvariables is not directly available. This nonlinear objective function is obtained from the flow field by the flow solver. We present and interpret numerical re sults of this implementation. Moreover, a contribution is given to a survey and a distinction of DFO research directions, to an analysis and discussion of these. We also state a derivative free algorithm used within a clustering algorithm in combination with non-smooth optimization techniques to reveal the effective ness of derivative free methods in computations. This algorithm is applied on some data sets from various sources of public life and medicine. We compare various methods, their practical backgrounds, and conclude with a summary and outlook. This work may serve as a preparation of possible future research. Keywords: Derivative Free Optimization, Quadratic Interpolation, Trust-Region Method, Well-Poisedness, Nonlinear Optimization, Stirrer Configurations, Support Vector Machines, Non-Smooth Optimization, Inverse Problems, Statistical Learning, Data, Clustering.