J Glob Optim DOI 10.1007/s10898-012-9861-z Higher-order optimality conditions in set-valued optimization using radial sets and radial derivatives Nguyen Le Hoang Anh · Phan Quoc Khanh Received: 11 July 2011 / Accepted: 24 January 2012 © Springer Science+Business Media, LLC. 2012 Abstract We propose higher-order radial sets and corresponding derivatives of a set-valued map and prove calculus rules for sums and compositions, which are followed by direct appli- cations in discussing optimality conditions for several particular optimization problems. Our main results are both necessary and sufficient higher-order conditions for weak efficiency in a general set-valued vector optimization problem without any convexity assumptions. Many examples are provided to explain advantages of our results over a number of existing ones in the literature. Keywords Higher-order radial sets · Higher-order radial derivatives · Calculus rules · Set-valued vector optimization · Weak efficiency · Higher-order optimality conditions Mathematics Subject Classification (2010) 90C46 · 49J52 · 46G05 · 90C26 · 90C29 1 Introduction Optimality conditions in nonsmooth problems have been attracting increasing efforts of mathematicians around the world for half a century. For systematic expositions about this topic, including practical applications, see books [3, 5, 24, 25, 27] and a recent survey [6]. A significant number of generalized derivatives have been introduced for studying optimality conditions in nonsmooth optimization. One can roughly separate the wide range of methods for nonsmooth problems into two groups: the primal space and the dual space approaches. N. L. H. Anh (B ) Department of Mathematics, University of Science of Hochiminh City, 227 Nguyen Van Cu, District 5, Hochiminh City, Vietnam e-mail: nlhanh@hcmus.edu.vn P. Q. Khanh Department of Mathematics, International University of Hochiminh City, Linh Trung, Thu Duc, Hochiminh City, Vietnam e-mail: pqkhanh@hcmiu.edu.vn 123