S13 - Real-World Applications II

We address the challenge of high-dimensional black-box optimization (BBO), where the objective function is unknown, non-differentiable, and expensive to evaluate. This problem arises in various real-world engineering tasks such as turbomachinery design using computational fluid dynamics (CFD) and automotive structural optimization, where hundreds of design variables must be tuned under physical and operational constraints.
While conventional BBO techniques-including Bayesian optimization, evolutionary strategies, and surrogate-assisted approaches-perform well in low-dimensional spaces, their effectiveness diminishes significantly as the number of decision variables increases. This limitation, often called the "curse of dimensionality," poses a significant challenge for high-dimensional BBO tasks in industry. Also, these methods do not natively handle constraints among the decision variables.
A BBO method based on quantum/quantum-inspired optimization solvers (QA-BBO) can address these issues associated with the conventional BBO methods. QA-BBO includes methods such as Factorization Machine with Quadratic-optimization Annealing (FMQA) [1, 2] and polynomial-based Kernels with Quadratic-optimization Annealing (Kernel-QA) [3]. QA-BBO utilizes surrogate models such as FM and kernel models within a serial optimization framework (like Bayesian optimization). While these surrogate models are typically low-dimensional, they require relatively a small number of model parameters, thereby mitigating overfitting even with small training datasets. 
In this work, we focus on two practical enhancements for QA-BBO-exponential transformation of objective values and training-data subsampling-and examine their impact on both benchmark and industrial problems.

Black-box optimization (BBO) is crucial in engineering domains (e.g., aerodynamic shape design), where the objective function is evaluated through computationally expensive simulations or physical experiments. Because these evaluations are time-consuming, the total number of evaluations becomes a critical bottleneck. Thus, efficient optimization methods that minimize the number of evaluations are essential.
Surrogate-based methods, such as Bayesian optimization [1], aim to reduce costly evaluations by constructing an approximate model of the objective and iteratively selecting promising candidates. Within this context, Factorization machine with quadratic optimization annealing (FMQA) [2, 3] has been proposed as a promising approach that couples a factorization-machine surrogate with an Ising-machine based combinatorial search.
A key limitation of the conventional FM [4] surrogate is its inability to capture the field structure of input variables. Although each original design variable (e.g., a categorical or discretized continuous parameter) naturally forms a "field" consisting of its one-hot encoded bits, conventional FM treats all these bits independently. This leads to inefficient learning when binary-encoded variables are used. To address this, we propose a field-aware extension of FMQA by replacing the conventional FM surrogate with a Field-Weighted Factorization Machine (FwFM) [5], resulting in the FwFMQA framework.
By combining a more expressive surrogate with efficient Ising-based search, FwFMQA significantly improves optimization efficiency in real-world applications where evaluation cost is dominant. Many practical problems involve categorical or discrete design choices such as material composition (selection of alloy families), photonic device layout (choice of lattice structures), drug-candidate scaffolds, and aerospace component configurations-where each variable naturally forms a field of one-hot encoded bits.
In addition, continuous design variables can be incorporated by discretizing them into binary representations. While this strategy is also employed in conventional FMQA, the proposed FwFMQA more effectively captures interactions among discretized variables through its field-aware structure. By treating each original discrete or discretized continuous variable as a field, the proposed method seamlessly handles both types of variables within the same framework. This unified treatment enables more accurate surrogate modeling and provides better guidance for the Ising-based search, resulting in more efficient optimization than conventional FMQA for black-box problems.

Combinatorial optimization problems refer to finding the optimal or near-optimal solution(s) within certain constraints or conditions. They are used in various fields, such as optimizing delivery routes in logistics and creating schedules. The characteristic feature of Ising machines is that they approach problems from the perspective of minimizing "energy". The energy function (Hamiltonian) of the Ising model can incorporate the individual states of the spins and their interactions. Ising machines-hardware solvers inspired by the Ising spin model-minimize an energy function (Hamiltonian) that captures both local terms and pairwise interactions. These include quantum annealers such as D-Wave, and quantum-inspired solvers leveraging GPUs or FPGAs. Furthermore, applying the quantum approximate optimization algorithm (QAOA) makes it possible to handle combinatorial optimization problems with quantum gate machines.
Ising machines that can take spin interactions into account can quickly solve quadratic un-constrained binary optimization (QUBO) problems, which are quadratic binary optimization problems (combinatorial optimization problems). In other words, the various optimization problems mentioned above can be solved by formulating them as quadratic polynomials based on binary variables. 
While constraints can be embedded in the QUBO formulation using penalty functions, determining suitable penalty weights (or suitable scaling for objectives to balance the penalty values) is non-trivial, especially in multi-objective optimization. Improper balancing between objectives and penalties can result in infeasible solutions. This challenge is more emphasized in practical applications, where the number of decision variables, objective functions, and constraints can be large and heterogeneous. Therefore, a systematic and scalable method for estimating objective scaling factors is critical to ensure effective optimization using quantum or quantum-inspired solvers.