Implementation of interior point methods for mixed semidefinite and second order cone optimization problems

Abstract

There is a large number of implementational choices to be made for the primal-dual interior point method in the context of mixed semidefinite and second order cone optimization. This paper presents such implementational issues in a unified framework, and compares the choices made by different research groups. This is also the first paper to provide an elaborate discussion of the implementation in SeDuMi.

A homogeneous interior-point algorithm for nonsymmetric convex conic optimization

“…See e.g. [1,23] for implementations of such fast algorithms for the case of self-scaled cones. Furthermore, in industry-standard software, heuristic techniques to speed up convergence rates are often employed, although they invalidate the proofs of convergence in the purely theoretical sense.…”

“…The Mehotra second order correction [12] is known to significantly improve practical performance of ipms for linear and quadratic conic problems [2,12,23]. With the same goal in mind, we suggest a new way to compute a search direction containing second order information for the general (possibly non-self-scaled) conic problem.…”

Conic Optimization Software

“…PENSDP [3], a modified version of PENNON is the only general purpose semidefinite programming solver using a different approach. The implementation of IPMs for conic optimization is more complicated than that for linear optimization, see [4][5][6] for more details.…”

“…It is one of the few solvers that used a self-dual embedding instead of an infeasible scheme to initialize the problem. It implements most of the techniques discussed in [5].…”

Magnetic resonance tissue quantification using optimal bSSFP pulse-sequence design

“…It is known that mixed SDO and SOCO problems can be solved efficiently with interior-point methods (IPMs), (see (Sturm, 2002)). Therefore, for the eigenvalue problem (NL-SDO) we introduce SDO-SOCO subproblems defined on a trust region.…”