On Squared-Variable Formulations for Nonlinear Semidefinite programming
Lijun Ding, Stephen J. Wright
TL;DR
This work shows that squared-variable reformulations of PSD-constrained problems preserve the essential second-order landscape: every 2NP of the original semidefinite program corresponds to a 2NP of the squared-variable reformulation, and vice versa, under mild conditions. It also analyzes a symmetric-F variant, where an eigenvalue condition guarantees the same correspondences for 2NC and local minimizers, clarifying when rotational symmetries impede strict equivalences. The results extend the applicability of equality-constrained optimization techniques to PSD-constrained settings and provide insights into the limits of local minimizer transfers, with implications for low-rank and nuclear-norm regularized problems. Overall, the paper contributes a matrix-analog of classical squared-variable ideas, highlighting when and how squared-slack formulations yield robust second-order optimality relationships and practical computational avenues.
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 $FF^\top$, 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.