Distance To Ill-posedness Of Conic Linear Optimization Problems

This thesis mainly concerned with characterizations and properties of distance to ill-posedness of conic linear optimization problems. When the conic linear op-timization problem is feasible, we first prove the distance to ill-posedness is the distance to infeasibility. Moreover, we present two mathematical programming, and then prove that their optimal value are identical, and equal to the distance to ill-posedness. Furtherly, we give the equivalent characterizations of the first program and point that the optimal value of the second program is exactly the geometric quantity defined by Belloni-Freund. When the conic linear optimization problem is infeasible, we prove the distance to ill-posedness is the distance to feasibility. We also give two kinds of mathematical programming, and then prove that their optimal value are identical, and equal to the distance to ill-posedness. In addition, we point that the width of the cone is the coefficient of linearity for the dual cone. Hence we present a mathematical programming to characterize the the distance to ill-posedness by its optimal value.Furthermore, we study the generalized conic linear optimization problems. Give sufficient and necessary condition of the feasibility of such a system, and then present equivalent characterization of the "distance to ill-posedness" for convex process. What's more, we generalized the results obtained above to the generalized conic linear optimization problem in the more general case.