Abstract
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.
Highlights
Since the groundbreaking paper of Karmarkar, many researchers have proposed and analyzed various interior-point methods (IPMs) for linear optimization (LO) and a large amount of results have been reported [ – ]
), which almost closes the gap between the iteration bounds for large- and small-update methods
The resulting iteration bounds for a wide class of eligible kernel functions have been outlined in a series of papers [, ] starting with [ ] for LO, we immediately get the iteration bounds for large- and small-update methods for convex quadratic optimization over symmetric cone (CQSCO)
Summary
Since the groundbreaking paper of Karmarkar, many researchers have proposed and analyzed various interior-point methods (IPMs) for linear optimization (LO) and a large amount of results have been reported [ – ]. In [ ], with x being the vector in Rr consisting of all the eigenvalues of the symmetric cone v This gives the following theorem, which yields a lower bound on δ(v) in terms of (v). We will show that the barrier function (v) in each inner iteration with the default step size α , as defined by ( ), is decreasing. ( ), we have the following theorem, which shows that the default step size ( ) yields a sufficient decrease of the barrier function value during each inner iteration. The following lemma provides an estimate for the number of inner iterations between two successive barrier parameter updates, in terms of and the parameters β and γ. –θ ε which means that the total number of iterations is completely determined by the parameters θ , β, γ , τ , and the eligible kernel function ψ(t)
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