Rigorous results in combinatorial optimization

Abstract

Many current deterministic solvers for NP-hard combinatorial optimization problems are based on nonlinear relaxation techniques that use floating point arithmetic. Occasionally, due to solving these relaxations, rounding errors may produce erroneous results, although the deterministic algorithm should compute the exact solution in a finite number of steps. This may occur especially if the relaxations are illconditioned or ill-posed, and if Slater’s constraint qualifications fail. We show how verified results can be obtained by rigorously bounding the optimal value of nonlinear semidefinite relaxations, even in the ill-posed case. All rounding errors due to floating point arithmetic are taken into account.