Primal-Dual Interior-Point Methods for Self-Scaled Cones

Abstract

In this paper we continue the development of a theoretical foundation for efficient primal-dual interior-point algorithms for convex programming problems expressed in conic form, when the cone and its associated barrier are self-scaled (see [NT97]). The class of problems under consideration includes linear programming, semidefinite programming and convex quadratically constrained quadratic programming problems. For such problems we introduce a new definition of affine-scaling and centering directions. We present efficiency estimates for several symmetric primal-dual methods that can loosely be classified as path-following methods. Because of the special properties of these cones and barriers, two of our algorithms can take steps that go typically a large fraction of the way to the boundary of the feasible region, rather than being confined to a ball of unit radius in the local norm defined by the Hessian of the barrier.

An inexact interior point method for L 1-regularized sparse covariance selection

“…In this paper, we choose P = W −1/2 where W 0 is the Nesterov-Todd (NT) scaling matrix satisfying W ZW = X for given X, Z ∈ S p ++ [24]. It has been shown in [46] that for X, Z ∈ S p ++ and our choice of P , H P (XZ) = γI if and only if XZ = γI.…”

“…When solving examples with linear constraints, (5), we use ANS method with its default updating parameter r ρ = 2. For the PPA method, the stopping condition used is similar to that in (24), and the tolerance is set to be Tol = 10 −6 . [7] to construct a random sparse inverse covariance matrix.…”

“…Some properties of the symmetrized Kronecker product can be found in the Appendix of [24]. We use · to denote the Frobenius norm for a matrix or Euclidean norm for a vector, and · 2 to denote the spectral norm of a matrix.…”

On the Convergence of Central Path and Generalized Proximal Point Method for Symmetric Cone Linear Programming

“…Nesterov and Todd [1] first proposed this optimization problem under the name of convex programming for self-scaled cones, and established polynomial complexity of primal-dual interior-point method applied to this problem using the so-called NT direction. It is well known that symmetric cone programming (SCP) includes linear programming (LP), semidefinite programming (SDP) and second order cone programming (SOCP) as special cases.…”

“…Lemma2.1. [ [18],Theorem13] For any x = ∑ r j=1 λ j (x)c j , let g sc be defined by (1). Then g sc is (continuously) differentiable at x if and only if g is (continuously) differentiable at all λ j (x).…”