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Tez (doktora) - Anadolu Üniversitesi, Anadolu Üniversitesi, Fen Bilimleri Enstitüsü, Matematik Anabilim Dalı, Kayıt no: 1035534, Pozitif homojen fonksiyonlar için Küçük ve ark. [41] tarafından geliştirilen zayıf alt/üst exhauster kavramları zayıf subdiferansiyel/süperdiferansiyel ve exhausterlar arasındaki ilişkiler yardımıyla kurulmuş ve exhausterların özel bir sınıfını oluşturmuşlardır. Aynı çalışmada, zayıf exhausterların kolaylıkla hesaplanabilmesi için Minkowski toplam ve fark işlemleriyle bazı geometrik yöntemler verilmiştir. Zayıf exhausterlar kullanılarak gerekli ve yeterli optimallik koşulları da verilmiştir.Ayrıca, zayıf alt (üst) exhausterların, zayıf subdiferansiyelin (zayıf süperdiferansiyelin) sadece sınır noktalarıyla indekslenerek indirgenebileceği gösterilmiştir [42]. İndirgenmiş zayıf exhausterlarla da optimallik koşulları ifade edilmiştir. Buçalışmada, (zayıf) eşlenik fonksiyon ile (zayıf) exhausterlar arasındaki ilişkiler incelenmiştir. İlk olarak, bir fonksiyonun zayıf subdiferansiyeli bu fonksiyonun zayıf eşleniği cinsinden ifade edilmiş ve bu yeni ifade zayıf alt exhausterların da zayıf eşlenik dönüşümle karakterize edilmesini sağlamıştır. Bu karakterizasyon kullanılarak, maksimizasyon problemleri için yeni optimallik koşulları elde edilmiştir. Daha sonra, konveks bir optimizasyon probleminin amaç fonksiyonunun yönlü türeviniamaç fonksiyonu kabul eden yeni bir optimizasyon problemi oluşturulmuştur. Bu problemin eşlenik dualinin çözümleri, asıl problemin değer fonksiyonunun üst exhausterına ait kümeler cinsinden ifade edilmiştir. Bununla birlikte, konveks olmayan optimizasyon problemleri de ele alınarak, bunların zayıf Fenchel dual problemlerinin çözümleri zayıf üst exhausterlarla karakterize edilmiştir. Böylelikle, her iki durumda da asıl problemin kritik noktalarını belirleyen yöntemler oluşturulmuştur. Kuasidiferansiyellenebilir fonksiyonların da minimizasyonu incelenerek amaç fonksiyonu kuasidiferansiyellenebilir olan bir optimizasyon probleminin, dc-fonksiyon olan yönlü türevini amaç fonksiyonu kabul eden bir DC programlama problemi kurulmuştur. Bu problemin ve DC dualinin çözüm kümeleri için karakterizasyonlar verilmiştir. Ayrıca, bu tip optimizasyon problemlerinin zayıf Fenchel dualinin çözümü için de bir karakterizasyon verilmiştir.

WOS: 000439579600021, Vectorization is a technique that replaces a set-valued optimization problem with a vector optimization problem. In this work, by using an extension of the Gerstewitz function, a vectorizing function is defined to replace a given set-valued optimization problem with respect to the set less order relation. Some properties of this function are studied. Moreover, relationships between a set-valued optimization problem and a vector optimization problem, derived via vectorization of this set-valued optimization problem, are examined. Furthermore, necessary and sufficient optimality conditions are presented without any convexity assumption.

WOS: 000435964300009, In this study, some new order relations on family of sets are introduced by using Minkowski difference. The relations between these orders and the ordering cone of the vector space are obtained. It is shown that depending on the corresponding cone, these order relations are partial orders on the family of nonempty bounded sets. Some relationships between these order relations and upper and lower set less order relations are investigated. Also, two scalarizing functions are introduced in order to replace set optimization problems with respect to these partial order relations with scalar optimization problems. Moreover, necessary and sufficient optimality conditions are presented.

WOS: 000415280800009, In this study, some optimality conditions for DCH-functions are given in terms of -faces using DC-duality approach. We introduce some geometric characterizations for the solution set of DCH-minimization problems by means of exposed faces and Minkowski difference. Moreover, we prove that given conditions are independent of the choices of representatives of DCH-functions. Also, some of these conditions are employed to find inf-stationary points for a quasidifferentiable optimization problem (QDP), i.e., the directional derivative of the objective function is a DCH-function. Therefore, we present a condition to find stationary points of a QDP. We give some examples to illustrate obtained results.

WOS: 000382960700003, In this study, some relationships between quasidifferentiability and weakly subdifferentiability of a function are investigated, and some results on calculating the quasidifferential in terms of weak exhausters are presented. Moreover, under some assumptions weak subdifferentiability of a quasidifferentiable function is proved and the weak subdifferential is obtained by using the quasidifferential. Under some assumptions, it is proved that a positively homogeneous and lower semicontinuous function is quasidifferentiable. It is also shown that directional differentiability and weak subdifferentiability imply quasidifferentiability. Furthermore, a method to evaluate the quasidifferential is given by using reduced weak lower exhausters., Commission of Scientific Research Projects of Anadolu University [1201F018], This research was supported by Commission of Scientific Research Projects of Anadolu University [1201F018].

WOS: 000360382200001, In this work, the notion of weak superdifferential is presented. Some calculation rules are given to evaluate weak subdifferential and weak superdifferential of some classes of functions represented by support functions. Moreover, some methods are obtained to calculate weak subdifferential of convex functions. In addition, the concept of weak lower and weak upper exhausters of positively homogeneous functions are introduced by using weak subdifferential and weak superdifferential, respectively. In terms of weak exhausters, some optimality conditions are given to find local or global minimizers/maximizers of some classes of functions., Anadolu University Scientific Research Projects Commission [1201F018], This study was supported by Anadolu University Scientific Research Projects Commission [grant number 1201F018].

WOS: 000353814300001, In this work, it is proved that weak exhausters of a positively homogeneous function can be reduced if weak subdifferential/superdifferential can be represented as the sum of a subset of it and weak subdifferential of zero function. It is also shown that weak exhausters can be reduced if this subset is the set of minimal elements of weak subdifferential/superdifferential with respect to weak subdifferential of zero function. At the end, some optimality conditions are given using reduced weak exhausters., Anadolu University Scientific Research Projects Commission [1201F018], The authors would like to thank the editor and anonymous referees for their kind and helpful remarks and comments. This study was supported by Anadolu University Scientific Research Projects Commission under the grant no 1201F018.