Verification of semidefinite optimization problems with application to variational electronic structure calculation

Abstract

In this thesis we develop ideas of rigorous verification in optimization. Semidefinite programming (SDP) is reviewed as one of the fundamental types of convex optimization with a variety of applications in control theory, quantum chemistry, combinatorial optimization as well as many others. We show, how rigorous error bounds for the optimal value can be computed by carefully postprocessing the output of a semidefinite programming solver. All the errors due to the floating point arithmetic or illconditioning of the problems are considered. We also use interval arithmetic as a powerful tool to model uncertainties in the input data. In the context of this thesis a software package implementing the verification algorithms was developed. We provide detailed explanations and show how efficient routines can be designed to manage real life problems. Criteria for detecting infeasible semidefinite programs and issuing certificates of infeasibility are formulated. Examples and results for benchmark problems are included. Another important contribution is the verification of the electronic structure problems. There large semidefinite programs represent a reduced density matrix variational method. Our algorithms allow the calculation of a rigorous lower bound for the ground state energy. The obtained results and modified algorithms are also of importance because they show how much we can benefit in terms of problem complexity from exploiting the specific problem structure.