| With the application of fuzzy optimization in many areas, researchers intend to extend more methods for solving classical optimization problems to fuzzy optimization. But it is very difficult because of the difference between operators in classical mathematics and those in fuzzy mathematics. This is the motivation for this paper to construct an algebra structure that has the commonness of fuzzy spaces and ordinary linear spaces. The work would provide a basic tool for the further study of fuzzy optimization, especially the algorithms that are similar to those of classical optimization problems. Based on the theory of the new algebra structure, two fuzzy optimization models (fuzzy scalar optimization and optimization with fuzzy relation constraints) are investigated in this paper. In what follows, we will introduce the main results of the paper briefly.The second chapter is devoted to the study of two new notions, i.e., quasi-domains and semi-linear spaces. Some basic concepts and properties of the notions are given. Firstly, the bases of semi-linear spaces are defined. The transformations on semi-linear spaces, the matrices defined on quasi-domains and the relationship between them are investigated. Secondly, the definition of proper semi-linear spaces is given and a kind of partial order in semi-linear spaces is derived by the operator in the spaces. Lastly, convex sets in semi-linear space, convex and concave mappings are introduced and some basic properties of them are presented. These new concepts are the base of the new fuzzy optimization model given in the next chapter.In the third chapter, based on the theory of semi-linear spaces, fuzzy sets, fuzzy linear spaces and the bases of fuzzy linear spaces are redefined from a point of view of fuzzy points. The new definition of fuzzy bases unifies the three given definitions that are different from one another in form. The fuzzy matrices defined on fuzzy scalar quasi-domain are also studied in the chapter. It is proved that fuzzy linear transformations can be characterized by fuzzy matrices mentioned above. According to the theory of semi-linear spaces, the partial order in fuzzy scalar quasi-domain is derived by the operator in the quasi-domain. And then, a new definition of fuzzy convex sets is given. Based on these works, a new fuzzy optimization model is proposed. The existence of the optimal solution of the problem is investigated. It is proved that if the objective function is a lower bounded fuzzy concave mapping in the feasible domain (defined in this paper), then the optimal solution of the problem must exist and the optimal value can be attained at somevertex of the feasible domain (defined in this paper). For the ordinary case, μ-certain solution is defined and an algorithm for obtaining μ-certain solutions is given. A kind of fuzzy metric is given based on the operators in fuzzy linear spaces at the end of the chapter and the completeness of fuzzy metric spaces is investigated.The last chapter focuses on a class of optimization problems with fuzzy relation in-equality(FRI) constraints. First of all, the feasible domain of the problem is described according to the theory of semi-linear spaces. Some new notions are defined, such as I-convex sets, I-convex polyhedrons, vertices and so on. And then, based on these new notions, we get a similar conclusion with that of linear programming. Based on the results obtained by Wang [69], a more general model of latticized linear programming is investigated as a special case of FRCO (optimization problems with fuzzy relation inequality constraints). By the minimal points choose principle given in this paper, we give a modified algorithm for the FRCO problem with a linear objective function. The new algorithm can get the optimal point even by one step for some problems. At last, we give an algorithm for the FRCO problems with smooth objective functions. Through some examples, it can be seen that this algorithm is better than the one given by Lu from a point of view of computation time.
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