| In this thesis,radius of robust global error bounds for an uncertain piecewise linear inequality system are investigated.And some characterizations of robust weakly efficient solutions for an sum of squares convex(SOS-convex)polynomial optimization problem with uncertain data are also investigated.The thesis is divided into four chapters.Chapter 1,at first,the research background of robust solutions is recalled.Then,the development and researches on the topic of error bounds for inequality systems and robust solutions for SOS-convex polynomial optimization problems are reviewed.Finally,the motivations and the main research work of this thesis are listed.Chapter 2 deals with the radius of robust global error bounds for an uncertain piecewise linear inequality systems under the polytope uncertain sets.Following the robust optimization,we first introduce the robust counterpart for this uncertain piecewise linear inequality systems.Then,we obtain a dual characterization for the robust global error bounds of the uncertain piecewise linear inequality system.Moreover,we establish upper and lower bounds for the radius of robust global error bounds of the system of uncertain piecewise linear inequalities in terms of the Minkowski function generalized by the polytope uncertain sets.Our results extend and improve the corresponding results in the literature.Chapter 3 is concerned with robust weakly efficient solutions for an uncertain SOS-convex polynomial optimization problem with spectrahedral uncertain data in both the objective and constraints.By using a robust type characteristic cone constraint qualification,we first obtain optimality conditions for robust weakly efficient solutions of this uncertain SOS-convex polynomial optimization problem in terms of sum of squares conditions and linear matrix inequalities.Then,we propose a relaxation dual problem for this uncertain SOS-convex polynomial optimization problem and explore weak and strong duality properties between them.Moreover,we give a numerical example to show that the relaxation problem can be reformulated as a semidefinite linear programming problem.Chapter 4 summarizes the main results of this thesis and gives some remaining questions in the future. |
