On Squared-Variable Formulations for Nonlinear Semidefinite programming

TL;DR

This paper investigates squared-variable reformulations in nonlinear semidefinite programming, analyzing how first- and second-order optimality conditions relate between the original and reformulated problems.

Contribution

It provides a detailed analysis of the relationships between optimality conditions for the original and squared-variable formulations in nonlinear semidefinite programming.

Findings

First-order points in squared-variable formulation do not necessarily correspond to original problem's first-order points.

Second-order points show closer correspondence between the reformulation and original problem.

Insights into local minimizers' relationships between the two formulations.

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.