J Syst Sci Complex (2011) 24: 394–400 GENERALIZED INVEXITY-TYPE CONDITIONS IN CONSTRAINED OPTIMIZATION Shashi K. MISHRA · Norma G. RUEDA DOI: 10.1007/s11424-011-8234-x Received: 18 June 2008 / Revised: 23 July 2009 c The Editorial Office of JSSC & Springer-Verlag Berlin Heidelberg 2011 Abstract This paper defines a new class of generalized type I functions, and obtains Kuhn-Tucker necessary and sufficient conditions and duality results for constrained optimization problems in the presence of the aforesaid weaker assumptions on the objective and constraint functions involved in the problem. Key words Constrained optimization, duality results, type I functions. 1 Introduction Hanson [1] introduced the concept of invexity in constrained optimization as a generalization of convexity. Later, Craven and Glover [2] proved that any differentiable scalar function is invex if and only if every stationary point is a global minimizer. Therefore, in the case of uncon- strained optimization, the concept of invexity is equivalent to the above property. However, in constrained optimization the objective and the constraint functions are assumed to be invex with respect to the same vector function. Several authors have considered possible relaxations of the invexity requirement in order to obtain necessary and sufficient conditions for optimality and duality theorems. Hanson and Mond [3] introduced two new classes of functions called type I and type II functions, which are not only sufficient but are also necessary for optimality in the primal and dual problems, respectively. Hanson [4] defined weak type I objective and constraint functions to solve the van der Waerden Permanent Problem, devised in 1926. For a comprehensive survey of generalized convexity with applications to duality theory and optimality conditions, the readers may refer to [5]. For a more recent publication on the subject, they may consult the book by Mishra and Giorgi [6] . Consider the following nonlinear programming problem: min f (x) s.t. g(x) ≤ 0. Shashi K. MISHRA Department of Mathematics, Faculty of Science, Banaras Hindu University, Varanasi 221005, India. Norma G. RUEDA Department of Mathematics, Merrimack College, North Andover MA 01845, USA. ⋄ This paper was recommended for publication by Editor Shouyang WANG.