Lyapunov-type Programming Over Symmetric Cones

The Lyapunov-type programming problem over symmetric cone is a special class of symmetric cone optimization problems and contains the Lyapunov-type program-ming over cone of semidefinite matrices as a specific case. It has wide applications in many fields such as optimal control, stability analysis and engineering design. In this thesis, by employing Euclidean Jordan algebraic technique, together with properties of Lyapunov operator and its generalized inverse, we consider the feasibility and solvabil-ity of the Lyapunov-type linear programming over symmetric cone, and the solution existence and uniqueness of the Lyapunov-type least-squares problem over symmet-ric cone. Moreover, we design two classes of interior-point methods for more general linear complementarity problems over symmetric cones, which henceforth provide the theoretical framework of interior-point algorithms for the Lyapunov-type programming problems.There are five chapters in this thesis.In Chapter 1, we give a brief introduction to the background, motivation and sig-nificance of Lyapunov-type programming over symmetric cone, summarize the main results of this thesis, review and develop some preliminaries on Euclidean Jordan alge-bra and its symmetric cone, define the generalized inverse of the Lyapunov operator and characterize some useful properties of Lyapunov operator and its generalized inverse.In Chapter 2, we consider the feasibility and solvability of the Lyapunov-type linear programming over symmetric cone. After establishing the models of the corresponding primal and dual problems, we propose two kinds of Lyapunov-type Farkas'lemmas, based on the properties of Lyapunov operator and its generalized inverse, to exhibit the desired feasibility of these two problems. We further show that the feasibilities of the primal and dual problems lead to the solvability of the primal problem and zero duality gap under some mild conditions. In this case, we obtain that any solution to the pair of primal and dual problems is equivalent to the one of the corresponding KKT system.In Chapter 3, we introduce a least-squares problem for the constraint system in the aforementioned Lyapunov-type linear programming and hence call it the Lyapunov-type least-squares problem over symmetric cone. Applying the tools in convex anal-ysis, decomposition techniques in Euclidean Jordan algebra, as well as properties of the image of symmetric cone under Lyapunov operator, we exploit the existence and uniqueness of such least-squares solution.In Chapter 4, we provide two classes of new interior-point methods over symmetric cones and establish their global convergence and complexity, respectively. By defining a new non-monotonicity, the Cartesian P*(k)-property, for linear operators, we design a class of polynomial-time interior-point algorithms with feasible initial point for the cor-responding non-monotone linear complementarity problem over symmetric cone. Note that the feasible initial point may be difficult to achieve. To make up such deficiency in this method, we also propose a new class of polynomial-time infeasible interior-point algorithm which is essentially a combination of the standard interior-point method and the smoothing Newton method. It is known that both the linear programming and the quadratic programming can be reformulated to some special linear complementarity problems. In this regard, the aforementioned two class of interior-point algorithms are definitely provide the theoretical framework of algorithms for two kinds of Lyapunov-type programming problems studied in Chapters 2 and 3.In Chapter 5, the main contributions of this thesis are concluded and some future research issues are presented.