Feasibility-driven QAOA with penalty scheduling

Abstract

Most available quantum algorithms address constrained optimization problems by treating constraints as soft penalty terms within a QUBO formulation. This approach requires careful adjustment of the penalty coefficients, which scales poorly with the number of constraints and lacks a proper strategy to balance feasibility and solution quality. In this work, we introduce two extensions of standard linear-ramp QAOA (lr-QAOA) tailored to problems with multiple heterogeneous constraints. We first construct $\Lambda$-lr-QAOA, in which each penalty term is assigned its own linear-ramp schedule, promoting penalty weights from external hyperparameters to internal variational parameters of QAOA, similarly to the objective and mixer parameters. By optimizing all schedules jointly in a single run, this approach eliminates nested penalty tuning and scales more efficiently to multiple constraints. The optimization is guided by a feasibility-driven loss function that pushes the quantum state towards high-quality feasible solutions. As a further refinement, we introduce piecewise-ramp QAOA, in which the linear ramps are replaced by two-segment piecewise schedules, enhancing the expressiveness of the Ansatz at the cost of a small parameter overhead independent of the circuit depth. We benchmark both methods on Earth-observation satellite mission planning tasks formulated as budget-constrained Maximum Weight Independent Set problems. Numerical results show that piecewise-ramp QAOA consistently outperforms lr-QAOA and $\Lambda$-lr-QAOA across circuit depths and system sizes. Furthermore, both $\Lambda$-lr-QAOA and piecewise-ramp QAOA exhibit a high feasibility rate, which is crucial in industrial applications. Our analysis highlights an intrinsic feasibility-optimality trade-off, which we address by introducing a filtered variant of the loss providing a single hyperparameter to tune this balance.