A paradox in bosonic energy computations via semidefinite programming relaxations

We show that the recent hierarchy of semidefinite programming relaxations based on non-commutative polynomial optimization and reduced density matrix variational methods exhibits an interesting paradox when applied to the bosonic case: even though it can be rigorously proven that the hierarchy collapses after the first step, numerical implementations of higher order steps generate a sequence of improving lower bounds that converges to the optimal solution. We analyze this effect and compare it with similar behavior observed in implementations of semidefinite programming relaxations for commutative polynomial minimization. We conclude that the method converges due to the rounding errors occurring during the execution of the numerical program, and show that convergence is lost as soon as computer precision is incremented. We support this conclusion by proving that for any element p of a Weyl algebra which is non-negative in the Schrodinger representation there exists another element p' arbitrarily close to p that admits a sum of squares decomposition.Comment: 22 pages, 4 figure

Strange Behaviors of Interior-point Methods for Solving Semidefinite Programming Problems in Polynomial Optimization

We observe that in a simple one-dimensional polynomial optimization problem (POP), the `optimal' values of semidefinite programming (SDP) relaxation problems reported by the standard SDP solvers converge to the optimal value of the POP, while the true optimal values of SDP relaxation problems are strictly and significantly less than that value. Some pieces of circumstantial evidences for the strange behaviours of SDP solvers are given. This result gives a warning to users of SDP relaxation method for POP to be careful in believing the results of the SDP solvers. We also demonstrate how SDPA-GMP, a multiple precision SDP solver developed by one of the authors, can deal with this situation correctly

Strange Behaviors of Interior-point Methods for Solving Semidefinite Programming Problems in Polynomial Optimization

We observe that in a simple one-dimensional polynomial optimization problem (POP), the `optimal' values of semidefinite programming (SDP) relaxation problems reported by the standard SDP solvers converge to the optimal value of the POP, while the true optimal values of SDP relaxation problems are strictly and significantly less than that value. Some pieces of circumstantial evidences for the strange behaviours of SDP solvers are given. This result gives a warning to users of SDP relaxation method for POP to be careful in believing the results of the SDP solvers. We also demonstrate how SDPA-GMP, a multiple precision SDP solver developed by one of the authors, can deal with this situation correctly