COMPUTER AIDED OPTIMIZATION TECHNIQUES BASED ON GENERALIZED CONVEX MAPS Gabriela Cristescu 1 , Mihail Găianu 2 1 “Aurel Vlaicu” University of Arad, România Department of Mathematics and Computers, gcristescu@inext.ro 2 West University of Timişoara Department of Computer Science, gaianumihail@yahoo.com Key words: convex set, convex function, g-convex function, level set, multiple criteria optimization, sequential approximation Abstract. Many times in multiple criteria optimization problems, the non-smoothness of the objective functions and of the admissible domain preserves some properties, which are enough to guarantee the existence of the solution. They are included in the rich class of of invexity properties, which bring useful properties in connection with approaching the vectorial optimization problems with generalized convex objective functions on generalized convex domain. The study was inspired by some multiple criteria optimization problem arising during modeling an ecologic-economic efficiency problem of a Romanian railway modernization project. A procedure of geometrical representation of the objective function by maps leads to a sequential approximation of the solution. A method of estimating the error is given. Many times in multiple criteria optimization problems, the non-smoothness of the objective functions and of the admissible domain preserves some properties, which are enough to guarantee the existence of the solution. As an example, the classes of g-convex sets and g-convex functions are described in this paper. The g-convexity is a particular type of invexity (see [4], [7], [9]), which brings useful properties in connection with approaching the vectorial optimization problems with g-convex objective functions on g- convex domain. The study was inspired by some multiple criteria optimization problem arising during modeling an ecologic-economic efficiency problem of a modernization project in the Romanian railway network. A procedure of geometrical representation of the objective function by maps leads to a sequential approximation of the solution. A method of estimating the error is given. First of all, let us remind that a set is convex whenever it contains the straight-line segment determined by each pair of its points. If (V, +, ·) is a real linear space then f: V  R is said to be a convex function if        y f x f y x f          1 1 whenever x, y are in V and  [0,1]. The roots of the idea lie in the book [5], in which the generalized convexity may be defined eventually by "replacing the linear structure of the space by another type of structure and defining a notion of straight-line segment". These kinds of convexity properties are called segmental convexities. As examples, are the concepts of • E-convexity defined by E. A. Youness [12] in R n , in which the straight line segment is <x,y>  λE(x)+(1-λ)E(y), E: R n R n . • g-convexity defined by M. A. Noor [10] in R n , in which the straight line segment is <x,y>  λx+(1-λ)g(y), g: R n  R n . • order convexity defined by G. Birkhoff in 1948 (see [5]), which bases on the order segment. • metric convexity defined by K. Menger in 1928 (see [5]), which bases on the order segment. • average convexities induced by metric structures (see [5]) ANNALS of the ORADEA UNIVERSITY. Fascicle of Management and Technological Engineering, Volume X (XX), 2011, NR1 3.34