A T-Algebraic Approach to Primal-Dual Interior-Point Algorithms

Abstract

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...

Existence and Uniqueness of Solutions for Homogeneous Cone Complementarity Problems

“…where the second inequality holds by the fact a 2 , e i = aa * , e i ≥ 0 for all a ∈ H because a = a * ; the last inequality follows from the Lipschitz continuity of F and ab ≤ a b for any a, b ∈ A by Theorem 2 of [11]. By Propositions 2.1 and 2.2,…”

“…Hence, by (10), x 0 = 0, a contradiction. If F has the P 0 property, then F has the P property, and hence by (11), x 0 = 0. This is also a contradiction.…”

“…, r} such that l ≥ j. (11) If F has the order-P 0 property, then F has the order-P property. Hence, by (10), x 0 = 0, a contradiction.…”

“…We first briefly review some basic concepts and results on homogeneous cones and T -algebras from [8,11,22], and then provide some fundamental results on metric projection onto homogenous cones.…”

“…While there is a finite number of nonisomorphic symmetric cones of each dimension, the number is uncountable for homogeneous cones when the dimension n is greater than 11; see [8]. There has been some increase in the interest and activity in the area of homogeneous cones and optimization problems over homogeneous cones; see, e.g., [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. These papers deal with either certain theoretical properties of homogeneous cones, primal-dual interior-point methods for linear optimization over homogeneous cones or their applications.…”