ON HAZAN’S ALGORITHM FOR SYMMETRIC PROGRAMMING PROBLEMS L. FAYBUSOVICH * Abstract. We describe the generalization of Hazan’s algorithm for symmetric programming problems Key words. Symmetric programming, Euclidean Jordan algebras, low-rank approximations to optimal solutions AMS subject classifications. 90C25,17C99 1. Introduction. In [3] E.Hazan proposed an algorithm for solving semidefinite programming problems. Though the complexity estimates for this algorithm are not as good as for the most popular primal-dual algorithms, it nevertheless has a number of interesting properties. It provides a low-rank approximation to the optimal solution and it requires the computation of the gradient of an auxiliary convex function rather than the Hessian of a barrier function in case of interior-point algorithms. In present paper we show that these properties are preserved in a much more general case of symmetric programming problems. Moreover, we will also show that the major com- putational step of the algorithm (finding the maximal eigenvalue and corresponding eigenvector of a positive definite symmetric matrix) admits a natural decomposition in a general case of a direct sum of simple Euclidean Jordan algebras. For some types of irreducible blocks (e.g. ,generally speaking, infinite-dimensional spin-factors) the corresponding eigenvalue problem has a simple analytic solution. In particular, Hazan’s algorithm offers a computationally attractive alternative to other methods in case of second-order cone programming (including the infinite-dimensional version considered in [2], [4]). 2. Jordan-algebraic concepts. We stick to the notation of an excellent book [1]. We do not attempt to describe the Jordan-algebraic language here but instead provide detailed references to [1]. Throughout this paper: • V is an Euclidean Jordan algebra; • rank(V ) stands for the rank of V ; • x ◦ y is the Jordan algebraic multiplication for x, y ∈ V ; • < x, y >= tr(x ◦ y) is the canonical scalar product in V ; here tr is the trace operator on V ; • Ω is the cone of invertible squares in V ; • ¯ Ω is the closure of Ω in V ; • An element g ∈ V such that g 2 = g and tr(g) = 1 is called a primitive idempotent in V ; • Given x ∈ V , we denote by L(x) the corresponding multiplication operator on V , i.e. L(x)y = x ◦ y, y ∈ V ; • Given x ∈ V , we denote by P (x) the so-called quadratic representation of x, i.e. P (x)=2L(x) 2 − L(x 2 ). * The author was supported in part by National Science Foundation Grant NO. DMS07- 12809. Department of Mathematics University of Notre Dame, Notre Dame, IN 46556 USA 1