The authors show transition points in parametric semidefinite optimization are finite and give a numerical algebraic geometry algorithm to find nonlinearity intervals and transition points.
On the local stability of semidefinite relaxations
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abstract
We consider a parametric family of quadratically constrained quadratic programs (QCQP) and their associated semidefinite programming (SDP) relaxations. Given a nominal value of the parameter at which the SDP relaxation is exact, we study conditions (and quantitative bounds) under which the relaxation will continue to be exact as the parameter moves in a neighborhood around the nominal value. Our framework captures a wide array of statistical estimation problems including tensor principal component analysis, rotation synchronization, orthogonal Procrustes, camera triangulation and resectioning, essential matrix estimation, system identification, and approximate GCD. Our results can also be used to analyze the stability of SOS relaxations of general polynomial optimization problems.
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On computing the nonlinearity interval in parametric semidefinite optimization
The authors show transition points in parametric semidefinite optimization are finite and give a numerical algebraic geometry algorithm to find nonlinearity intervals and transition points.