Kernel-function Based Algorithms for Semidefinite Optimization

Abstract

Abstract. Recently, Y.Q. Bai, M. El Ghami and C. Roos [3] introduced a new class of so-called eligible kernel functions which are defined by some simple conditions. The authors designed primal-dual interiorpoint methods for linear optimization (LO) based on eligible kernel functions and simplified the analysis of these methods considerably. In this paper we consider the semidefinite optimization (SDO) problem and we generalize the aforementioned results for LO to SDO. The iteration bounds obtained are analogous to the results in [3] for LO.

Kernel function with BFGS quasi-newton methods for solving nonlinear semi-definite problems

“…[19] introduced a primal-dual IPM specifically designed for SDO, achieving complexity results of O(q 2 √ n log n ε ), and O( √ n log n log n ε ) for updates at both small and large scales, with q > 1. In [12] extended the IPM from LO to SDO and obtained similar iteration bounds.…”

Complexity of primal-dual interior-point algorithm for linear programming based on a new class of kernel functions

“…They presented a unified analysis of primal-dual IPMs based on eligible KFs for LP. The results obtained in [6] were successfully extended to SDP, CQP, convex quadratic semidefinite programming (CQSDP), SOCP, convex quadratic programming over symmetric cone (SCCQP), P * (κ)−linear complementarity problem (LCP), the Cartesian P * (κ)−SCLCP and symmetric programming in [9,13,16,20,37,52,53,55] respectively.…”

A unified kernel function approach to primal-dual interior-point algorithms for convex quadratic SDO

“…Bai et al [3][4][5][6][7][8] proposed a class of primal-dual IPMs for LO based on a variety of non-self-regular kernel functions and obtained the same favorable iteration bounds for large-and small-update methods as in [21]. Moreover, Bai and her co-authors extended the aforementioned results for LO to second-order cone optimization (SOCO) [9], SDO [10,26] and CQSDO [27], respectively. For some other related interior-point algorithms based on the kernel functions we refer to [1,11,15,20].…”