Solving rank-constrained semidefinite programs in exact arithmetic

TL;DR

This paper introduces an exact symbolic algorithm for solving rank-constrained semidefinite programs, providing a new approach that is efficient under certain conditions and applicable to various fields like control theory and polynomial optimization.

Contribution

The paper develops a symbolic, exact algorithm for rank-constrained semidefinite programs with quadratic complexity, outperforming numerical methods in certain cases.

Findings

Algorithm is polynomial in the number of variables for fixed matrix size.

Complexity is quadratic relative to natural degree bounds.

Implementation in Maple demonstrates practical viability.

Abstract

We consider the problem of minimizing a linear function over an affine section of the cone of positive semidefinite matrices, with the additional constraint that the feasible matrix has prescribed rank. When the rank constraint is active, this is a non-convex optimization problem, otherwise it is a semidefinite program. Both find numerous applications especially in systems control theory and combinatorial optimization, but even in more general contexts such as polynomial optimization or real algebra. While numerical algorithms exist for solving this problem, such as interior-point or Newton-like algorithms, in this paper we propose an approach based on symbolic computation. We design an exact algorithm for solving rank-constrained semidefinite programs, whose complexity is essentially quadratic on natural degree bounds associated to the given optimization problem: for subfamilies of the problem where the size of the feasible matrix is fixed, the complexity is polynomial in the number of variables. The algorithm works under assumptions on the input data: we prove that these assumptions are generically satisfied. We also implement it in Maple and discuss practical experiments.