Projection methods for conic feasibility problems: applications to polynomial sum-of-squares decompositions

Abstract

This paper presents a projection-based approach for solving conic feasibility problems. To find a point in the intersection of a cone and an affine subspace, we simply project a point onto this intersection. This projection is computed by dual algorithms operating a sequence of projections onto the cone and generalizing the alternating projection method. We release an easy-to-use Matlab package implementing an elementary dual-projection algorithm. Numerical experiments show that, for solving some semidefinite feasibility problems, the package is competitive with sophisticated conic programming software. We also provide a particular treatment for semidefinite feasibility problems modelling polynomial sum-of-squares decomposition problems.

Projection Methods in Conic Optimization

“…The final section presents some numerical experiments with regularization methods on polynomial optimization problems, showing the interest of the approach in that context. This chapter is meant to be an elementary presentation of parts of the material of several papers; among those, our main references are [Mal04], [QS06], [MPRW09], [HM11], [ZST10] and [Nie09]. We aim at clarifying the ideas, presenting them in a general framework, unifying notation, and most of all, pointing out what makes things work.…”

“…In fact, this assumption yields moreover that there exists solutions to (17) (note that the assumption has also a natural geometrical appeal in context of projection methods, see [HM11,Sec. 3]).…”

“…corresponds exactly to the alternating projection method (9) (see [Mal04] for a proof in the special case of correlation matrices, and [HM11] for the proof in general). We thus have a (surprizing) dual interpretation of the primal projection method.…”

“…Since then, this dual method has been used successfully to solve real-life projection problems in numerical finance (among them: the problem of calibrating covariance matrices in robust portfolio selection [Mal04, 5.4]). A simple Matlab implementation has been made publicly available together with [HM11] for pedagogical and diffusion purposes.…”

“…Section 4.2 focuses on semidefinite feasibility problems arising when testing positivity of polynomials. We refer to the introduction of [HM11] for more examples and references.…”