Benchmarking of quantum and classical SDP relaxations for QUBO formulations of real-world logistics problems

TL;DR

This work benchmarks SDP relaxations of $QUBO$ reformulations for real-world logistics problems (OVRP and ASP), comparing SOS-based SDP methods with Hamiltonian Updates under practical, industry-derived data. It demonstrates that non-generic instances pose substantial challenges, with structure-exploiting solvers (notably sparsity-aware SOS variants) delivering the strongest lower bounds on ASP problems, while HU offers scalable bounds but with large gaps on these instances. The findings underscore the importance of exploiting problem structure and careful precision choices, and they reveal that quantum-inspired methods currently do not outperform classical SDP approaches on these real-world benchmarks. Overall, the study highlights the need for tailored preprocessing and solver development to close the gap between SDP relaxations and exact IP/IQP solutions in industry-scale problems.

Abstract

Quadratic unconstrained binary optimization problems (QUBOs) are intensively discussed in the realm of quantum computing and polynomial optimization. We provide a vast experimental study of semidefinite programming (SDP) relaxations of QUBOs using sums of squares methods and on Hamiltonian Updates. We test on QUBO reformulations of industry-based instances of the (open) vehicle routing problem and the (affinity-based) slotting problem -- two common combinatorial optimization problems in logistics. Beyond comparing the performance of various methods and software, our results reaffirm that optimizing over non-generic, real-world instances provides additional challenges. In consequence, this study underscores recent developments towards structure exploitation and specialized solver development for the used methods and simultaneously shows that further research is necessary in this direction both on the classical and the quantum side.

Figures (8)

• Figure 1: Visualization of the pipeline for this work: We compare different methods and software for semidefinite programming (SDP) relaxations of quadratic unconstrained binary optimization (QUBO) formulations that emerge from real-life applications. In particular, we consider the open vehicle routing problem (OVRP) and the affinity-based slotting problem (ASP). The applications are initially formulated as integer program (IP) or integer quadratic program (IQP) with either a linear objective function (for OVRP) or a quadratic objective function (for ASP), respectively, that can be solved to optimality.
• Figure 2: Sparsity and distribution of non-zero entries of the matrix corresponding to QUBO ASP30-250. All values are scaled by dividing with the largest entry of absolute value. The largest values by absolute value lie on the diagonal, this is indicated by different colors on the diagonal.
• Figure 3: Objective values$Z_{\text{QUBO}}$ for solving the QUBO formulations of ASP30 with different penalties using CPLEX and a time limit of 120 seconds. The red line indicates the optimal value $Z^*$ of ASP30. The blue dots are the objective values $Z_{\text{QUBO}}$, additionally the red stars indicate that the optimal value was reached for these formulations.
• Figure 4: Relative Difference $\Delta_{\text{SOS}}^{\text{rel}}$ on all ASP30 instances with different penalties, obtained with first order relaxations, $2d=2$.
• Figure 5: Relative Difference $\Delta_{\text{SOS}}^{\text{rel}}$ on all ASP30 instances with different penalties, obtained with TSSOS, using Mosek with the second order relaxation, $2d=4$ and the default tolerances of $e-8$, as well as $e-4$.