2025/02/04 by Ding, Lijun, Wright, Stephen J. · 1 citation
#FOS: Mathematics #Optimization and Control (math.OC)
paper · doi:10.48550/arxiv.2502.02099
In optimization problems involving smooth functions and real and matrix variables, that contain matrix semidefiniteness constraints, consider the following change of variables: Replace the positive semidefinite matrix X ∈ \mathbbSd, where \mathbbSd is the set of symmetric matrices in ℝd× d, by a matrix product FF^\top, where F ∈ ℝd × d or F ∈ \mathbbSd. The formulation obtained in this way is termed ``squared variable," by analogy with a similar idea that has been proposed for real (scalar) variables. It is well known that points satisfying first-order conditions for the squared-variable reformulation do not necessarily yield first-order points for the original problem. There are closer correspondences between second-order points for the squared-variable reformulation and the original formulation. These are explored in this paper, along with correspondences between local minimizers of the two formulations.