Complexity, condition numbers, and conic linear systems

The unifying theme of this thesis is the study of measures of conditioning for convex feasibility problems in conic linear form, or conic linear systems. Such problems are an important tool in mathematical programming. They provide a very general format for studying the feasible regions of convex optimization problems (in fact, any convex feasibility problem can be modeled as a conic linear system), and include linear programming feasibility problems as a special case. Over the last decade many important developments in linear programming, most notably, the theory of interior-point methods, have been extended to convex problems in this form. In recent years, a new and powerful theory of “condition numbers” for convex optimization has been developed. The condition numbers for convex optimization capture the intuitive notion of problem “conditioning” and have been shown to be important in studying the efficiency of algorithms, including interior-point algorithms, for convex optimization as well as other behavioral characteristics of these problems such as geometry, etc. The contribution of this thesis is twofold. We continue the research in the theory of condition numbers for convex problems by developing an elementary algorithm for solving a conic linear system, whose complexity depends on the condition number of the problem. We also discuss some potential drawbacks in using the condition number as the sole measure of conditioning of a conic linear system, motivating the study of “data-independent” measures. We introduce a new measure of conditioning for feasible conic linear systems in special form and study its relationship to the condition number and other measures of conditioning arising in recent linear programming literature. We study many of the implications of the new measure for problem geometry, conditioning, and algorithm complexity, and demonstrate that the new measure is data-independent. We also introduce the notion of “pre-conditioning” for conic linear systems, i.e., finding equivalent formulations of the problem with better condition numbers. We characterize the best such formulation and provide an algorithm for constructing a formulation whose condition number is within a known factor of the best possible. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690.)