The Radius Of Robust Feasibility For Uncertain Convex Optimization Problems

This thesis is devoted to study the radius of the robust feasibility for uncertain convex optimization problems with convex polynomial constraints,support vector machine problems with uncertain data,and adjustable robust linear programs with affine decision rules.This thesis is divided into five chapters.In Chapter 1,the research background of uncertain convex optimization problems is first described.Then,the development and researches on the topic of the radius of robust feasibility for uncertain convex optimization problems are introduced.Finally,the motivation and the content of this thesis are given.In Chapter 2,we deal with the lower bound of the radius of the robust feasibility for a class of uncertain convex optimization problems with convex polynomial constraints.Following the idea due to robust optimization,we first introduce the robust counterpart of the uncertain convex optimization problem and give the concept of radius of robust feasibility.By using the so-called epigraphical set and the Minkowski functions generated by the uncertain sets,we obtain the lower bound for the radius of robust feasibility of the uncertain convex optimization problem.Furthermore,an exact formula for the radius of the robust feasibility for an uncertain optimization problem with SOS-convex polynomial constraints is obtained.In Chapter 3,we are concerned with the robust feasibility for a class of support vector machine problems with uncertain data.Firstly,a robust counterpart problem of the uncertain support vector machine problem is introduced in terms of robust optimization.Then,an equivalent problem of the robust counterpart problem of the uncertain support vector machine problem is given.Finally,by using the equivalent problem and the socalled epigraphical set,an exact formula for the radius of robust feasibility of the uncertain support vector machine problem is obtained.In Chapter 4,we deal with the radius of robust feasibility for an adjustable robust linear program with affine decision rules.To do this,we first present a linear program reformulation for this adjustable robust linear program.Then,in terms of this linear program reformulation,we establish computable upper and lower bounds for the radius of robust feasibility for this adjustable robust linear program.Finally,we obtain an exact formula for calculating the radius of robust feasibility for a general parametric linear homogeneous system with adjustable variables.In Chapter 5,the main results of this thesis are summarized,and then some remaining questions which will be considered in the future are addressed.