A Framework for Automatically Setting Multiple Penalty Weights in Quadratic Unconstrained Binary Optimization

This paper proposes a novel framework for automatically setting penalty weights in the Quadratic Unconstrained Binary Optimization (QUBO) to handle problems with multiple constraints. Quantum computing and quantum-inspired methods have gained prominence, with QUBO models widely used for their unified representation of optimization problems. However, QUBO performance is highly sensitive to penalty weight settings, especially in multiple constraints scenarios, where setting different penalty weights to different constraints may further affect performance. Existing methods for setting common penalty weights for all constraints in QUBO can be generally categorized into manual methods based on expert experience, exact methods based on the objective function values, and sequential methods based on searching within an upper bound. This study introduces a novel framework for multiple constraints optimization that utilizes exact and sequential binary search methods to determine common penalty weight while integrating constraint sensitivity analysis to adjust each penalty weight dynamically. This approach enhances the efficiency and quality of QUBO solutions. Experimental results on the Travelling Salesman Problem show that the proposed framework achieves superior solution quality and feasibility on both the D-Wave quantum annealer and classical simulated annealing solvers, outperforming traditional methods and demonstrating strong adaptability.