General penalty-weight prescription for higher-order polynomial optimization

Determine a generally applicable prescription for choosing an appropriate penalty weight in higher-order polynomial unconstrained binary optimization problems, including cubic and quartic cluster-expansion Hamiltonians with composition constraints.

Background

The paper optimizes cubic and quartic cluster-expansion Hamiltonians for Au–Cu alloys by adding a quadratic composition-penalty term to the higher-order polynomial objective. Choosing the penalty weight is essential: a weight that is too small permits composition violations, whereas a weight that is too large can dominate the physical energy objective and hinder the search for low-energy configurations.

The authors use a derivative-based heuristic, calibrated empirically for the systems and solvers studied. They explicitly state that no generally applicable rule is currently available for selecting penalty weights in higher-order polynomial problems, leaving open the development of a principled prescription that can transfer across Hamiltonians, constraints, and optimization solvers.

References

However, for higher-order polynomial problems, there is still no generally applicable prescription for choosing an appropriate penalty weight.

— Constrained Optimization of Higher-Order Cluster-Expansion Hamiltonians for Alloys Using Simulated Bifurcation  (2609.10310 - Ichikawa et al., 9 Sep 2026) in Section 3.2, “Estimation of Penalty Weight”