Dual certificates of primal cone membership
Joonyeob Lee, D\'avid Papp, Anita Varga

TL;DR
This paper introduces a new framework for efficiently certifying membership in convex cones using dual certificates, enabling practical verification and optimization in polynomial and cone programming.
Contribution
It develops verifiable primal membership certificates from dual cone vectors assuming a self-concordant barrier, applicable to various polynomial cones and low-dimensional optimization.
Findings
Certificates can be computed using interior-point methods.
Applicable to cones like SOS, SONC, SAGE, SDSOS in polynomial optimization.
Enables direct optimization over low-dimensional dual cones.
Abstract
We discuss optimization problems over convex cones in which membership is difficult to verify directly. In the standard theory of duality, vectors in the dual cone are associated with separating hyperplanes and interpreted as certificates of non-membership in the primal cone . Complementing this perspective, we develop easily verifiable certificates of membership in given by vectors in . Assuming that admits an efficiently computable logarithmically homogeneous self-concordant barrier, every vector in the interior of is associated with a full-dimensional cone of efficiently verifiable membership certificates. Consequently, rigorous certificates can be computed using numerical methods, including interior-point algorithms. The proposed framework is particularly well-suited to optimization over low-dimensional linear images of higher dimensional cones: we argue…
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Taxonomy
TopicsPoint processes and geometric inequalities · Advanced Algebra and Geometry
