Kernel-function-based primal-dual interior-point methods for convex quadratic optimization over symmetric cone

Abstract

In this paper, we give a unified computational scheme for the complexity analysis of kernel-function-based primal-dual interior-point methods for convex quadratic optimization over symmetric cone. By using Euclidean Jordan algebras, the currently best-known iteration bounds for large-and small-update methods are derived, namely, O( √ r log r log r ε ) and O( √ r log r ε ), respectively. Furthermore, this unifies the analysis for a wide class of conic optimization problems. MSC: 90C25; 90C51

A Parametric Kernel Function Generating the best Known Iteration Bound for Large-Update Methods for CQSDO

“…Since 0 < t < 1 and q ≥ 1, it follows that q 2 − q + 1 > 0 and −t 3 + qt 2 + q 3 ≥ (q − 1)t 3 + q 3 > 0. This proves (10). We furthermore have for all t > 1, and…”

“…Moreover, Bai and her co-authors extended the aforementioned results for LO to SDO [20] and CQSDO [19]. We note that similar algorithms are successfully prolonged to convex quadratic optimization over symmetric cone (CQSCO) (see [10,21]). For some other related interior-point algorithms based on the kernel functions we refer to [1,3,6,11,18,22,23,24,25].…”

“…It was also shown that (11) and 12imply (14). So to prove the eligibility, we verify only the conditions from (10) to (13).…”

“…The kernel function ψ(t) was called eligible if it satisfies (10) to (14). It was also shown that (11) and 12imply (14).…”

Primal-dual interior point QP-free algorithm for nonlinear constrained optimization

“…Later, under the assumption that the multiplier approximation sequence remains bounded, the PTH algorithm was improved by Gao et al [ 3 ] by solving an extra SLE. The PTH algorithm was also improved by Qi and Qi [ 23 ], Zhu [ 26 ] and Cai [ 28 ].…”