Relating homogeneous cones and positive definite cones via T-algebras Chua, Chek Beng. 2003 Chua, C. B., (2003). Relating homogeneous cones and positive definite cones via T-algebras.

Abstract. Three primal-dual interior-point algorithms for homogeneous cone programming are presented. They are a short-step algorithm, a large-update algorithm, and a predictor-corrector algorithm. These algorithms are described and analyzed based on a characterization of homogeneous cone via T -algebra. The analysis show that the algorithms have polynomial iteration complexity.Key words and phrases. Homogeneous cone programming; T-algebra; Primal-dual interior-point algorithm. IntroductionPrimal-dual interior-point algorithms-first designed for linear programming (see, e.g., [19]), and subsequently extended to semidefinite programming (see, e.g., [18, Part II]) and symmetric cone programming (see, e.g., [13])-are the most widely used interior-point algorithms. At the same time, they are able to achieve the best iteration complexity bound known to date. These strongly motivates further development of primal-dual interior-point algorithms for wider classes of optimization problems.Various studies to extend primal-dual algorithms beyond symmetric cone programming involves the v-space approach; i.e., the use of scaling points. Tunçel [15] showed that primal-dual symmetric algorithms that use the v-space approach can only be designed for symmetric cone programming. By dropping primal-dual symmetry, Tunçel [16] designed primal-dual algorithms based on the v-space approach. However, polynomial iteration complexity bounds were established only for symmetric cone programming. This paper focuses on primal-dual interior-point algorithms for homogeneous cone programming.Homogeneous cone programming, which will be formally defined in the next section, is a class of convex programming problems. It includes, as a proper sub-class, symmetric cone programming. While there is a finite number of non-isomorphic symmetric cones of each dimension, this number is uncountable for homogeneous cones when the dimension is at least eleven [17]. Thus the collection of homogeneous cones is significantly larger than the subclass of symmetric cones.On the other hand, the author [4] proved that every homogeneous cone can be represented as the intersection of some cone of symmetric positive definite matrices with a suitable linear subspace. Thus all homogeneous cone programming problems can be formulated as semidefinite programming problems, which are special cases of symmetric cone programming problems. However it is argued in the same paper that there are advantages in designing algorithms specifically for homogeneous cone programming as the ranks of these cones-which determines the iteration complexity of interior-point The primal-dual algorithms in this paper rely on a characterization of homogeneous cones via T -algebras.Algebraic structures of convex cones had been successfully exploited in the design and analysis of interior-point algorithms for convex conic programming. This is most notable for symmetric cone programming [1,3,8,14]) where Jordan algebraic structures of symmetric cones were used. The primal-dual algorithms and their anal...

In this paper, we introduce a new P -type property for nonlinear functions defined over Euclidean Jordan algebras, and study a continuation method for nonlinear complementarity problems over symmetric cones. This new P -type property represents a new class of nonmonotone nonlinear complementarity problems that can be solved numerically.

The notion of weighted centers is essential in V-space interior-point algorithms for linear programming. Although there were some successes in generalizing this notion to semidefinite programming via weighted center equations, we still do not have a generalization that preserves two important properties-(1) each choice of weights uniquely determines a pair of primal-dual weighted centers, and (2) the set of all primal-dual weighted centers completely fills up the relative interior of the primal-dual feasible region. This paper presents a new notion of weighted centers for semidefinite programming that possesses both uniqueness and completeness. Furthermore, it is shown that under strict complementarity, these weighted centers converge to weighted centers of optimal faces. Finally, this convergence result is applied to homogeneous cone programming, where the central paths defined by a certain class of optimal barriers for homogeneous cones are shown to converge to analytic centers of optimal faces in the presence of strictly complementary solutions.

We consider two notions for the representations of convex cones: G-representation and lifted-G-representation. The former represents a convex cone as a slice of another; the latter allows in addition, the usage of auxiliary variables in the representation. We first study the basic properties of these representations. We show that some basic properties of convex cones are invariant under one notion of representation but not the other. In particular, we prove that lifted-G-representation is closed under duality when the representing cone is self-dual. We also prove that strict complementarity of a convex optimization problem in conic form is preserved under G-representations. Then we move to study efficiency measures for representations. We evaluate the representations of homogeneous convex cones based on the "smoothness" of the transformations mapping the central path of the representation to the central path of the represented optimization problem.