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arXiv:2501.04971 · Design, Automation, and Test in Europe

Self-Adaptive Ising Machines for Constrained Optimization

Corentin Delacour

Published Mar 31, 2025• 1 citation
arXivHTMLPDF

Abstract

Ising machines (IM) are physics-inspired alternatives to von Neumann architectures for solving hard optimization tasks. By mapping binary variables to coupled Ising spins, IMs can naturally solve unconstrained combinatorial optimization problems such as finding maximum cuts in graphs. However, despite their importance in practical applications, constrained problems remain challenging to solve for IMs that require large quadratic energy penalties to ensure the correspondence between energy ground states and constrained optimal solutions. To relax this requirement, we propose a self-adaptive IM that iteratively shapes its energy landscape using a Lagrange relaxation of constraints and avoids prior tuning of penalties. Using a probabilistic-bit (p-bit) IM emulated in software, we benchmark our algorithm with multidimensional knapsack problems (MKP) and quadratic knapsack problems (QKP), the latter being an Ising problem with linear constraints. For QKP with 300 variables, the proposed algorithm finds better solutions than state-of-the-art IMs such as Fujitsu's Digital Annealer and requires 7,500x fewer samples. Our results show that adapting the energy landscape during the search can speed up IMs for constrained optimization.

DOI
10.23919/date64628.2025.10992982

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Ising machines (IM) are physics-inspired alternatives to von Neumann architectures for solving hard optimization tasks. By mapping binary variables to coupled Ising spins, IMs can naturally solve unconstrained combinatorial optimization problems such as finding maximum cuts in graphs. However, despite their importance in practical applications, constrained problems remain challenging to solve for IMs that require large quadratic energy penalties to ensure the correspondence between energy ground states and constrained optimal solutions. To relax this requirement, we propose a self-adaptive IM that iteratively shapes its energy landscape using a Lagrange relaxation of constraints and avoids prior tuning of penalties. Using a probabilistic-bit (p-bit) IM emulated in software, we benchmark our algorithm with multidimensional knapsack problems (MKP) and quadratic knapsack problems (QKP), the latter being an Ising problem with linear constraints. For QKP with 300 variables, the proposed algorithm finds better solutions than state-of-the-art IMs such as Fujitsu's Digital Annealer and requires 7,500x fewer samples. Our results show that adapting the energy landscape during the search can speed up IMs for constrained optimization.
Combinatorial optimization has wide applications from industry to natural science. Ising machines bring an emerging computing paradigm for efficiently solving a combinatorial optimization problem by searching a ground state of a given Ising model. Current cutting-edge Ising machines achieve fast sampling of near-optimal solutions of the max-cut problem. However, for problems with additional constraint conditions, their advantages have been hardly shown due to difficulties in handling the constraints. In this work, we focus on benchmarks of Ising machines on the quadratic knapsack problem (QKP). To bring out their practical performance, we propose fast two-stage post-processing for Ising machines, which makes handling the constraint easier. Simulation based on simulated annealing shows that the proposed method substantially improves the solving performance of Ising machines and the improvement is robust to a choice of encoding of the constraint condition. Through evaluation using an Ising machine called Amplify Annealing Engine, the proposed method is shown to dramatically improve its solving performance on the QKP. These results are a crucial step toward showing advantages of Ising machines on practical problems involving various constraint conditions.
The binary quadratic knapsack problem (QKP) aims at optimizing a quadratic cost function within a single knapsack. Its applications and difficulty make it appealing for various industrial fields. In this paper we present an efficient strategy to solve the problem by modeling it as an Ising spin model using an Ising machine to search for its ground state which translates to the optimal solution of the problem. Secondly, in order to facilitate the search, we propose a novel technique to visualize the landscape of the search and demonstrate how difficult it is to solve QKP on an Ising machine. Finally, we propose two software solution improvement algorithms to efficiently solve QKP on an Ising machine.
Physics-based Ising machines (IM) have been developed as dedicated processors for solving hard combinatorial optimization problems with higher speed and better energy efficiency. Generally, such systems employ local search heuristics to traverse energy landscapes in searching for optimal solutions. Here, we quantify and address some of the major challenges met by IMs by extending energy-landscape geometry visualization tools known as disconnectivity graphs. Using efficient sampling methods, we visually capture landscapes of problems having diverse structure and hardness manifesting as energetic and entropic barriers for IMs. We investigate energy barriers, local minima, and configuration space clustering effects caused by locality reduction methods when embedding combinatorial problems to the Ising hardware. To this end, we sample disconnectivity graphs of PUBO energy landscapes and their different QUBO mappings accounting for both local minima and saddle regions. We demonstrate that QUBO energy-landscape properties lead to the subpar performance for quadratic IMs and suggest directions for their improvement.
An Ising machine (IM), as a type of analog computer tailored for tackling intractable combinatorial optimization problems, has attracted remarkable attention in recent years. In contrast to the blossoming field of bespoke IM hardware, developing metaheuristics from IMs remains largely uninvestigated. Here, we propose a physics-inspired evolutionary computation paradigm, termed the Ising optimization framework (IOF); it comprises a unique Ising algorithm and a hybrid annealing scheme, which together are well-suited for solving quadratic unconstrained binary optimization (QUBO) problems embedded in Ising system energy. The Ising algorithm leverages a set of iterated self-mapping functions to evolve an Ising-spin swarm, enabling efficient energy minimization in artificial Ising systems while mitigating detrimental chaos. Complementing the algorithm, a hybrid annealing scheme integrating singular value dropout, bifurcation control, and a nudging strategy, is devised to augment the overall optimization capacity. The effectiveness of the IOF is validated on various Ising and Max-cut problems with decision variables ranging from 625 to 5000 in number. In comparison to four major types of methods for solving QUBOs, including IM simulations, nature-inspired algorithms, a state-of-the-art heuristic, and the commercial solver Gurobi, the IOF consistently demonstrates notable optimization quality and computational efficiency. This paper provides a theoretical foundation and practical guidelines for bridging Ising-inspired approaches with evolutionary computation, offering an evolutionary perspective on Ising optimizations and suggesting a fertile avenue for future research and application.
Ising machines are next-generation computers expected to efficiently sample near-optimal solutions of combinatorial optimization problems. Combinatorial optimization problems are modeled as quadratic unconstrained binary optimization (QUBO) problems to apply an Ising machine. However, current state-of-the-art Ising machines still often fail to output nearoptimal solutions due to the complicated energy landscape of QUBO problems. Furthermore, the physical implementation of Ising machines severely restricts the size of QUBO problems to be input as a result of limited hardware graph structures. In this study, we take a new approach to these challenges by injecting auxiliary penalties preserving the optimum, which reduces quadratic terms in QUBO objective functions. The process simultaneously simplifies the energy landscape of QUBO problems, allowing the search for near-optimal solutions, and makes QUBO problems sparser, facilitating encoding into Ising machines with restriction on the hardware graph structure. We propose linearization of QUBO problems using variable posets as an outcome of the approach. By applying the proposed method to synthetic QUBO instances and to multi-dimensional knapsack problems, we empirically validate the effects on enhancing minorembedding of QUBO problems and the performance of Ising machines.
The extended Ising machine (EIM) enhances conventional Ising models, which handle only binary quadratic forms by allowing constraints through real-valued dependent variables. We address the quadratic knapsack problem (QKP), hard to solve using Ising machines when formulated as a quadratic unconstrained binary optimization (QUBO). We demonstrated the EIM's superiority by comparing it with the conventional Ising model-based approach, a commercial exact solver, and a state-of-the-art heuristic solver for QKP.