A new full-Newton step O(n) infeasible interior-point algorithm for semidefinite optimization

Abstract

Interior-point methods for semidefinite optimization have been studied intensively, due to their polynomial complexity and practical efficiency. Recently, the second author designed an efficient primal-dual infeasible interior-point algorithm with full Newton steps for linear optimization problems. In this paper we extend the algorithm to semidefinite optimization. The algorithm constructs strictly feasible iterates for a sequence of perturbations of the given problem and its dual problem, close to their central paths. Two types of full-Newton steps are used, feasibility steps and (ordinary) centering steps, respectively. The algorithm starts from strictly feasible iterates of a perturbed pair, on its central path, and feasibility steps find strictly feasible iterates for the next perturbed pair. By using centering steps for the new perturbed pair, we obtain strictly feasible iterates close enough to the central path of the new perturbed pair. The starting point depends on a positive number ζ. The algorithm terminates in at most O n log n ε steps either by finding an ε-solution or by determining that the primal-dual problem pair has no optimal solution with vanishing duality gap satisfying a condition in terms of ζ.

Improved Complexity Analysis of Full Nesterov–Todd Step Feasible Interior-Point Method for Symmetric Optimization

“…The obtained iteration bounds coincide with the ones derived for LO, where n is replaced by r , the rank of EJAs, which are the currently best known iteration bounds for SO. Some other references, related to full step IPMs, without any attempt to be complete, are [12,[20][21][22][23][24].…”

Infeasible Interior-Point Methods for Linear Optimization Based on Large Neighborhood

“…A future research may focus on some new analysis of the algorithm which yields some larger value of θ . -Roos' full-Newton step IIPM was extended to semidefinite optimization (SDO) by Mansouri and Roos [23], to symmetric optimization (SO) by Gu et al [24] and to LCP by Mansouri et al [25]. An extension of large-update FIPMs based on kernel functions to SDO was presented by El Ghami [10].…”

A Modified Infeasible-Interior-Point Algorithm for Linear Optimization Problems

“…where r 0 b and r 0 c are the initial residual vectors as defined in (9). Note that, if ν = 1, then x = x 0 yields a strictly feasible solution of (P ν ), and (y, s) = (y 0 , s 0 ) a strictly feasible solution of (D ν ).…”

“…Later on, more variants of this algorithm were published. Mansouri et al [9,10] extended this algorithm on semidefinite optimization and linear complementarity problems (LCPs). Gu et al presented an improved IIPM for LO in [11].…”