Solving Rank-Constrained Semidefinite Programs in Exact Arithmetic

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, or the dimension of the affine section, is fixed, the algorithm is polynomial time. The algorithm works under assumptions on the input data: we prove that these assumptions are generically satisfied. We implement it in Maple and discuss practical experiments.

Solving generic nonarchimedean semidefinite programs using stochastic game algorithms

“…Moreover, as mentioned above, the computation of exact solutions of semidefinite programming problems is of current interest. In particular, Nie, Ranestad, and Sturmfels [NRS10] provided complexity measures based on the notion of algebraic degree, and dedicated algorithms have been developed by Henrion, Naldi, and Safey El Din [HNSED16,Nal18].…”

Mixed-Projection Conic Optimization: A New Paradigm for Modeling Rank Constraints

“…Among others, d 'Aspremont (2003) proposed reformulating low-rank constraints as systems of polynomial equations which can be addressed via the sum-of-squares hierarchy (Lasserre 2001). More recently, Naldi (2018) proposed a semi-algebraic reformulation of rank-constrained SDOs, which can be optimized over via Gröbner basis computation (Cox et al 2013). Unfortunately, algebraic approaches do not scale well in practice.…”

Mixed-Projection Conic Optimization: A New Paradigm for Modeling Rank Constraints

“…Unfortunately, while the first 1-2 levels of the hierarchy successfully supply high-quality bounds for low-rank problems, solving low-rank problems exactly requires the full hierarchy, which scales notoriously poorly in practice. More recently, Naldi (2018) proposed a semi-algebraic reformulation of rank-constrained SDOs, which can be optimized over via Gröbner basis computation (see Cox et al 2013, for a general theory). Unfortunately, their approach only scales to successfully solve low-rank problems where n = 5 and r = 3 in 1, 000s of seconds.…”