An equivalent nonlinear optimization model with triangular low-rank factorization for semidefinite programs

TL;DR

This paper introduces a new nonlinear optimization model for semidefinite programs using triangular low-rank matrices, which improves local optimality properties and reduces decision variable dimension.

Contribution

It presents a novel nonlinear model with triangular low-rank matrices that guarantees strict local optima and has a smaller decision variable space compared to previous models.

Findings

The new model guarantees the existence of strict local optima under certain conditions.

Numerical results show improved efficiency over existing models.

Reduced decision variable dimension enhances computational performance.

Abstract

In this paper, we propose a new nonlinear optimization model to solve semidefinite optimization problems (SDPs), providing some properties related to local optimal solutions. The proposed model is based on another nonlinear optimization model given by Burer and Monteiro (2003), but it has several nice properties not seen in the existing one. Firstly, the decision variable of the proposed model is a triangular low-rank matrix, and hence the dimension of its decision variable space is smaller. Secondly, the existence of a strict local optimum of the proposed model is guaranteed under some conditions, whereas the existing model has no strict local optimum. In other words, it is difficult to construct solution methods equipped with fast convergence using the existing model. Some numerical results are also presented to examine the efficiency of the proposed model.