An auto-balancing/scaling method for multi-objectives with constraints

Constraints in QUBO formulations are typically enforced through penalty terms with a unity penalty value. For example, a corresponding penalty function  may be used to enforce the constraint . In multi-objective optimization problems involving objectives and constraints with associated penalties , a key challenge lies in selecting appropriate scaling factors to balance the optimization landscape.

We propose a systematic three-step approach to estimate these scaling factors:

1. Constraint-only optimization: First, solve a simplified problem with only the constraints (no objectives are considered). All found  feasible solutions will satisfy .
2. Evaluation and averaging: Evaluate each objective function  over the feasible solutions, and compute the average to derive a scaling factor:

      .

Here,  is a hyper parameter and typically . 

1. Combined optimization: Use these scaling factors to normalize the objectives by , combine them with penalties, and solve the complete optimization problem.

This method provides a systematic way to balance conflicting terms in the QUBO, leading to higher-quality and feasible solutions, while also allowing for priority-weighting among objectives if needed.

To demonstrate the effectiveness of the proposed method, we applied it to an energy mix optimization problem in the context of a Home Energy Management System (HEMS). To minimize cost and environmental impact, HEMS dynamically selects energy sources-such as grid electricity, photovoltaic (PV) generation, battery storage, and electric vehicle (EV) discharging.

The formulated problem includes three objective functions: cost minimization, minimization of demands-supply difference, and battery wear minimization. It also includes three constraints: energy balance, capacity limits, and operational feasibility. By applying the proposed scaling factor estimation technique, we first identified feasible solutions via constraint-only optimization (step 1). Then, using these solutions, we computed averaged objective values to determine proper scaling factors (step 2) and solved the full optimization problem using those scaling factors (step 3).

We executed this process using a GPU-based simulated annealer. The solutions satisfied all constraints and achieved a well-balanced trade-off among objectives and constraints at different problem settings. The assessment validated the robustness and adaptability of our method across real-world multi-objective problem settings.

Real-world QUBO formulations frequently involve multiple heterogeneous objectives and constraints, where the relative scaling between objectives and penalty terms critically affects solution feasibility and quality. In practice, these scaling parameters are highly problem-dependent and may even change over time due to varying operating conditions, making manual tuning impractical and non-scalable.

The proposed methodology addresses this challenge by introducing a data-driven and systematic approach to estimate objective scaling factors based on the structure of the feasible solution space. By decoupling constraint satisfaction and objective evaluation, the method enables robust normalization of objectives without requiring prior domain-specific parameter tuning. This significantly reduces the barrier to applying QUBO-based optimization in large-scale, real-world systems.

Importantly, the approach is inherently scalable and compatible with a wide range of Ising-based solvers, including quantum annealers, quantum-inspired hardware, and GPU-based simulators. Furthermore, because the formulation remains within the QUBO/Ising framework, it can be naturally extended to gate-based quantum algorithms such as QAOA, supporting hybrid quantum-classical workflows.

This flexibility makes the method particularly suitable for dynamic optimization scenarios such as energy systems, logistics, and manufacturing, where problem structures evolve over time and robust, automated scaling is essential for sustained operational performance.