Optimality And Duality For Generalized Convex Uncertain Programming

Optimality and duality are important parts of the theory of optimization and have been studied for over a century, while the convexity plays an important role in the study of optimality conditions and dualities for optimization problems. In recent years, with the rapid development of uncertain programming, the studies of optimality conditions and dualities for uncertain programming have been made much attention from the researchers, and have became a hot research topic. The study of this paper focus on the optimality con-ditions and dualities of the problems of interval-valued programming, robust optimization, multiobjective variational control programming and the new models of robust optimization.The major works of this paper are summarized as follows:1. The Karush-Kuhn-Tucker optimality conditions and dualities are studied in a class of generalized convex optimization problems with an interval-valued objective function. Firstly, the concepts of preinvex and invex for real-valued functions are extended to interval-valued functions. Secondly, several properties of interval-valued preinvex functions and interval-valued invex functions are proposed; the Karush-Kuhn-Tucker optimality condi-tions are derived for the LU-preinvex and invex optimization problems with an interval-valued objective function under the conditions of weakly continuously differentiable and Hukuhara differentiable. Thirdly, the relationships between a class of variational-like in-equalities and the interval-valued optimization problems are established. Finally, based on the concepts of having no duality gap in weak and strong sense, the Wolfe duality theorems for the invex interval-valued nonlinear programming problems are proposed.2. For the Conservative of robust linear optimization model with the uncertainty data, a new robust linear optimization formulation is proposed. Through the introduction of the new distance formula, the uncertainty data is mapped to the unit ball in order to improve the robust linear optimization model. The new model overcomes the conservative of the original model when the data perturbations are larger, so the solution of the new model gets a relatively good balance between robustness and optimality. Several standard practical problems are tested; the simulation results show that the new model not only can ensure the robustness of solution but also has good optimality.3. For a class of G-invex nonlinear programming problems in the face of data un-certainty, robust G-Karush-Kuhn-Tucker necessary and sufficient optimality conditions are established; and the robust duality theorems between the primal problem and the Mond-Weir dual problem are presented, the results show that, the use of this paper that the conclusions on a more simple in calculation.4. For the problems of multiobjective variational programming and multiobjective variational control programming, firstly, the concept of vector-valued G-invexity, which is introduced by Antczak for multiobjective programming problems, is extended to generalized vector-valued G-type I invexity for multiobjective variational problems. By using Lagrange multiplier conditions, a number of sufficient optimality results are established under various types of generalized vector-valued G-type I invexity requirements, and duality results are also obtained for Mond-Weir type duals under above generalized vector-valued G-type I invexity assumptions. Secondly, the concepts of vector-valued G-invex functions are extended to multiobjective variational control problems. By using the new concepts, a number of sufficient optimality results and Mond-Weir type duality results are obtained for multiobjective variational control programming problems.