J Glob Optim (2011) 50:235–247 DOI 10.1007/s10898-010-9576-y On constructing total orders and solving vector optimization problems with total orders Mahide Küçük · Mustafa Soyertem · Yalçın Küçük Received: 5 February 2010 / Accepted: 22 June 2010 / Published online: 6 July 2010 © Springer Science+Business Media, LLC. 2010 Abstract In this paper, we introduce a construction method of total ordering cone on R n . It is shown that any total ordering cone on R n is isomorphic to the cone R n lex . Existence of a total ordering cone that contain given cone with a compact base is shown. By using this cone, a solving method of vector and set valued optimization problems is presented. Keywords Nonconvex analysis · Vector optimization · Total order · Nonconvex optimization · Scalarization 1 Introduction In vector optimization problems, the main purpose is to find optimal elements of a given set in partially ordered linear spaces. In the late 1800’s Edgeworth [1] and Pareto [2] provided the first definition of optimality, which is still being used in this field. Koopmans [3] and Kuhn-Tucker [4] made the first important mathematical contributions to the vector optimi- zation. Hurwicz [5], Vogel [6], Kirsch et al. [7] and Penot [8] are among those who studied vector optimization in partially ordered vector space. In [9, 10] recent developments on vector optimization can be found. Set-valued optimization is an extension of vector optimization to set-valued problems. The application of vector optimization principles to set-valued problems has received increasing M. Küçük (B ) · M. Soyertem · Y. Küçük Faculty of Science, Department of Mathematics, Anadolu University, Yunus Emre Campus, Eski¸ sehir, Turkey e-mail: mkucuk@anadolu.edu.tr M. Soyertem e-mail: soyertem@gmail.com Y. Küçük e-mail: ykucuk@anadolu.edu.tr 123