A new primal-dual path-following interior-point algorithm for semidefinite optimization

Abstract

In this paper we present a new primal-dual path-following interior-point algorithm for semidefinite optimization. The algorithm is based on a new technique for finding the search direction and the strategy of the central path. At each iteration, we use only full Nesterov-Todd step. Moreover, we obtain the currently best known iteration bound for the algorithm with small-update method, namely, O ( √ n log n ), which is as good as the linear analogue.

A Parametric Kernel Function Yielding the Best Known Iteration Bound of Interior-Point Methods for Semidefinite Optimization

“…In order to facilitate discussion, we denote : ( ), V δ δ = and we have the following result [8]. decreases sufficiently.…”

“…For the details we leave it for the interested readers (see, e.g., [8,15]. Following the strategy considered in [8], we briefly recall how to choose the default step size. Suppose that the step size α satisfies ( ( ) 2 ) ( ( )) 2 .…”

“…Many interior-point methods (IPMs) for linear optimization (LO) are successfully extended to (SDO) due to their polynomial complexity and practical efficiency. For an overview of these results, we refer to [1,2] and the references [3,4,5,6,7,8,9,10,11,12,13].…”

A full-Newton step feasible interior-point algorithm for P∗(κ)-LCP based on a new search direction

“…Achache [2], Asadi and Mansouri [4] and Kheirfam [12] presented numerical results on LCPs based on this technique. Later on, Achache [1], Wang and Bai [27,28,29] and Wang et al [30] extended Darvay's algorithm for LO to convex quadratic optimization (CQO), semidefinite optimization (SDO), second-order cone optimization (SOCO), symmetric cone optimization (SCO) and P * (κ)-LCP, respectively. Kheirfam introduced an infeasible IPM for SCO in [13].…”

An interior-point method for the Cartesian P*(k)-linear complementarity problem over symmetric cones

“…The search direction of his algorithm is introduced by using an algebraic equivalent transformation of the nonlinear equations which define the central path and then applying Newton's method for the new system of equations. Later on, Wang and Bai [15,16] extended Darvay's algorithm for LO to SDO and symmetric optimization (SCO). Recently, Zhang and Xu [21] proposed a full-Newton step primal-dual interior-point algorithm for LO.…”