Exact algorithms for determinantal varieties and semidefinite programming

Abstract

In this thesis we focus on the study of determinantal structures arising in semidefinite programming (SDP), the natural extension of linear programming to the cone of symetric positive semidefinite matrices. While the approximation of a solution of a semidefinite program can be computed efficiently by interior-point algorithms, neither efficient exact algorithms for SDP are available, nor a complete understanding of its theoretical complexity has been achieved. In order to contribute to this central question in convex optimization, we design an exact algorithm for deciding the feasibility of a linear matrix inequality (LMI) $A(x) \succeq 0$. When the spectrahedron $\spec = \{x \in \RR^n \mymid A(x) \succeq 0\}$ is not empty, the output of this algorithm is an algebraic representation of a finite set meeting $\spec$ in at least one point $x^*$: in this case, the point $x^*$ minimizes the rank of the pencil on the spectrahedron. The complexity is essentially quadratic in the degree of the output representation, which meets, experimentally, the algebraic degree of semidefinite programs associated to $A(x)$. This is a guarantee of optimality of this approach in the context of exact algorithms for LMI and SDP. Remarkably, the algorithm does not assume the presence of an interior point in the spectrahedron, and it takes advantage of the existence of low rank solutions of the LMI. In order to reach this main goal, we develop a systematic approach to determinantal varieties associated to linear matrices. Indeed, we prove that deciding the feasibility of a LMI can be performed by computing a sample set of real solutions of determinantal polynomial systems. We solve this problem by designing an exact algorithm for computing at least one point in each real connected component of the locus of rank defects of a pencil $A(x)$. This algorithm admits as input generic linear matrices but takes also advantage of additional structures, and its complexity improves the state of the art in computational real algebraic geometry. Finally, the algorithms developed in this thesis are implemented in a new Maple library called {Spectra}, and results of experiments highlighting the complexity gain are provided.

Locations

• HAL (Le Centre pour la Communication Scientifique Directe) - View - PDF