An improved and modified infeasible interior-point method for symmetric optimization

Abstract

In this paper an improved and modified version of full Nesterov–Todd step infeasible interior-point methods for symmetric optimization published in [A new infeasible interior-point method based on Darvay’s technique for symmetric optimization, Ann. Oper. Res. 211(1) (2013) 209–224; G. Gu, M. Zangiabadi and C. Roos, Full Nesterov–Todd step infeasible interior-point method for symmetric optimization, European J. Oper. Res. 214(3) (2011) 473–484; Simplified analysis of a full Nesterov–Todd step infeasible interior-point method for symmetric optimization, Asian-Eur. J. Math. 8(4) (2015) 1550071, 14 pp.] is considered. Each main iteration of our algorithm consisted of only a feasibility step, whereas in the earlier versions each iteration is composed of one feasibility step and several — at most three — centering steps. The algorithm finds an [Formula: see text]-solution of the underlying problem in polynomial-time and its iteration bound improves the earlier bounds factor from [Formula: see text] and [Formula: see text] to [Formula: see text]. Moreover, our method unifies the analysis for linear optimization, second-order cone optimization and semidefinite optimization.

A new search direction for full-Newton step infeasible interior-point method in linear optimization

“…The obtained complexity bound corresponds to the currently best-known theoretical iteration bound for IIPMs. In Table 1 we compare the obtained complexity results with the complexity bounds for IIPMs in [23,16,11,12,13,14]. Algorithm in [23] ( 1 5 , 1 8n )(n ≥ 2) 8n log max{nξ 2 ,∥r 0 b ∥,∥r 0 c ∥} ϵ Algorithm in [16] ( Algorithm in [11] ( 1 6(1+2κ) , 1 27n(1+2κ) 2 ) 27n(1 + 2κ) 2 log max{(x 0 ) T s 0 ,∥rq∥∥} ϵ…”

“…Roos [23] introduced an improved version of the method for LO that does not require centering steps, while the aforementioned methods require several (at most three) centering steps in each (main) iteration. Kheirfam extended this method to HLCP [11], the Cartesian P * (κ)-LCP [12], the convex quadratic symmetric cone optimization (CQSCO) [13] and SO [14]. Kheirfam [7] proposed an infeasible version of the method presented in [4] for SDLCP.…”

An infeasible full NT-step interior point method for circular optimization

“…Each main iteration of the aforementioned IIPMs is composed of one so-called feasibility step and a several centering steps to get an -optimal solution of the underlying problem. Recently, Roos [23] and Kheirfam [12,13,14] proposed IIPMs for LO, HLCP, the Cartesian P * (κ)-SCLCP and SCO so that their algorithms do not need centering steps and take only one feasibility step in order to get a new iterate close enough to the central path.…”

“…Motivated by Roos [23] and Kheirfam [12,13,14], we present a full-NT step IIPM for CO. Each main iteration of the proposed algorithm is consisted of only one feasibility step. Moreover, we analyze the algorithm and derive the iteration-complexity bound which matches the currently best-known iteration bound for IIPMs.…”

A new search direction for full-Newton step infeasible interior-point method in linear optimization

“…The author is improved this algorithm so that the algorithm performs only one feasibility step in each iteration and does not need centering steps [20]. Kheirfam [11,12,13,14] extended the algorithm proposed in [20] to HLCP, the Cartesian P * (κ)-LCP, the convex quadratic symmetric cone optimization (CQSCO) and SO. By considering the AET technique based on the function ψ(t) = t − √ t, Darvay et al [3] have introduced a full-Newton step IPM for LO.…”