On Squared-Variable Formulations for Nonlinear Semidefinite programming

Abstract

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 $X\in {\mathbb{S}}^d$, where ${\mathbb{S}}^d$ is the set of symmetric matrices in ${\mathbb{R}}^{d\times d}$, by a matrix product $F{F}^{\top }$, where $F\in {\mathbb{R}}^{d\times d}$ or $F\in {\mathbb{S}}^d$. 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.

34 pages