New complexity analysis of a full Nesterov–Todd step interior-point method for semidefinite optimization

Abstract

In this paper, we propose a new primal-dual path-following interior-point method for semidefinite optimization based on a new reformulation of the nonlinear equation of the system which defines the central path. The proposed algorithm takes only full Nesterov and Todd steps and therefore no line-searches are needed for generating the new iterations. The convergence of the algorithm is established and the complexity result coincides with the best-known iteration bound for semidefinite optimization problems.

A class of new search directions for full-NT step feasible interior point method in semidefinite optimization

“…To ensure the strict feasibility of the new iterates we used a factor ξ = 0.95. In our computational study, we compared our algorithm where ψ(t) = t p , p ≥ 2 with the variant of interior point algorithms that use the following AET for solving SDO problems: ψ(t) = t , ψ(t) = √ t and ψ(t) = t − √ t (see [7,11,19], respectively) where the value of lb is 1 4 . In all cases, the accuracy parameter had a value = 10 −5 .…”

“…In this paper, we use the Nesterov-Todd symmetrization scheme [2,7,10,11,17,[19][20][21], which defines the so-called NT-direction. Let us define the matrix…”

“…In the last decade, SDO has become a very active research area in mathematical programming because of the extension of the most algorithms for LO to the SDO case. Several primal-dual interior-point methods (IPMs) suggested for LO have been successfully extended to SDO [7,11,14,19], convex quadratic semidefinite optimization (CQSDO) [2,10] and other optimization problems [9,18,20,22] due to their polynomial complexity and practical efficiency. The first primal-dual feasible IPM with a full-Newton step for LO was proposed by Roos et al [16].…”

“…The strategy is based on an algebraic equivalent transformation (AET) of the standard centering equations of the central path ψ( xz µ ) = ψ(e) where ψ(t) = √ t. Achache [1], Wang and Bai [19,20], extended Darvay's algorithm for LO to convex quadratic optimization (CQO), SDO and second-order cone optimization (SOCO), respectively. In 2016, Darvay et al [5] developed a new full-Newton step feasible IPM for LO based on a new reformulation of the standard centering equations of the central path with ψ(t) = t − √ t. Kheirfam [11], generalized this method for SDO and derived the currently best-known iteration bound for SDO problems. In 2018, Darvay and Takács [6] designed a feasible primal-dual interior point algorithm for LO.…”