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Sums of increasing convex and increasing concave functions
Institution:1. Nanyang Business School, Nanyang Technological University, Singapore;2. School of Management, University of Science and Technology of China, China;1. National Institute for Public Health and the Environment, Bilthoven, The Netherlands;2. Department of Biological and Environmental Sciences and Technologies, University of Salento, Lecce, Italy;1. Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA;2. Departments of Computer Science and Mathematical Sciences, Kent State University, Kent, OH 44242, USA;3. College of Science, Technology, Engineering, Mathematics (STEM), Youngstown State University, Youngstown, OH 44555-3347, USA;4. Institute of Mathematics, Faculty of Mechanical Engineering, Brno University of Technology, Technicka 2896/2, 616 69 Brno, Czech Republic;1. School of Computer and Information Technology, Shanxi University, Taiyuan 030006, China;2. Key Laboratory of Computational Intelligence, Chinese Information Processing of Ministry of Education, Taiyuan 030006, China;3. School of Foreign Languages, Shanxi University, Taiyuan 030006, China
Abstract:Necessary and sufficient conditions for a function to be representable as a sum of an increasing convex and an increasing concave function are given. Adding a complementary slackness requirement yields a uniquely determined representation. The function class has applications to the theory of majorization and provides an instance of ‘best’ decomposition of functions which are differences of convex functions.
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