The cardinality constrained index-tracking problem seeks to find a combination of financial assets to replicate the return of a broad market index while using only a small subset of the asset universe. However, finding this combination of assets is NP-hard. In this thesis, we show the computational efficacy of solving a quadratic unconstrained binary optimization (QUBO) problem using purpose-built hardware Digital Annealer. We express an intra-cluster distance minimization problem as a QUBO problem and examine the computational performance between Digital Annealer and a commercial solver Gurobi running on conventional hardware. Moreover, we develop a new framework to solve the cardinality constrained index-tracking problem using the quadratic k-medoids clustering method on a market graph. To circumvent the nature of NP-hardness, we use the penalty method to formulate the problem as a QUBO problem, allowing us to utilize purpose-built hardware to solve the problem.
The cardinality constrained index-tracking problem seeks to find a combination of financial assets to replicate the return of a broad market index while using only a small subset of the asset universe. However, finding this combination of assets is NP-hard. In this thesis, we show the computational efficacy of solving a quadratic unconstrained binary optimization (QUBO) problem using purpose-built hardware Digital Annealer. We express an intra-cluster distance minimization problem as a QUBO problem and examine the computational performance between Digital Annealer and a commercial solver Gurobi running on conventional hardware. Moreover, we develop a new framework to solve the cardinality constrained index-tracking problem using the quadratic k-medoids clustering method on a market graph. To circumvent the nature of NP-hardness, we use the penalty method to formulate the problem as a QUBO problem, allowing us to utilize purpose-built hardware to solve the problem.
This article empirically examines the computational cost of solving a known hard problem, graph clustering, using novel purpose-built computer hardware. We express the graph clustering problem as an intra-cluster distance or dissimilarity minimization problem. We formulate our poblem as a quadratic unconstrained binary optimization problem and employ a novel computer architecture to obtain a numerical solution. Our starting point is a clustering formulation from the literature. This formulation is then converted to a quadratic unconstrained binary optimization formulation. Finally, we use a novel purpose-built computer architecture to obtain numerical solutions. For benchmarking purposes, we also compare computational performances to those obtained using a commercial solver, Gurobi, running on conventional hardware. Our initial results indicate the purpose-built hardware provides equivalent solutions to the commercial solver, but in a very small fraction of the time required.
A key strategy to enhance specialized hardware accelerators, such as GPUs and FPGA, is to reduce the numerical precision in arithmetic operations, which increases processing speed and lowers latency-both crucial for real-time AI applications. In this work, we consider NP-hard Quadratic Unconstrained Binary Optimization (QUBO) problems, which arise in machine learning and beyond. We show that these problems often require high numerical precision, posing challenges for hardware solvers. We introduce a principled Branch-andBound algorithm for reducing the precision requirements of QUBO problems by utilizing the dynamic range as measure of complexity. Experiments demonstrate that our method reduces the dynamic range in subset sum, clustering, and vector quantization problems, thereby increasing their solvability on actual quantum annealers and FPGA-based digital annealers.
Several optimization solvers inspired by quantum annealing have been recently developed, either running on actual quantum hardware or simulating it on traditional digital computers. Industry and academics look at their potential in solving hard combinatorial optimization problems. Formally, they provide heuristic solutions for Ising models, which are equivalent to quadratic unconstrained binary optimization (QUBO). Constraints on solutions feasibility need to be properly encoded. We experiment on different ways of performing such an encoding. As benchmark we consider the cardinality constrained quadratic knapsack problem (CQKP), a minimal extension of QUBO with one inequality and one equality constraint. We consider different strategies of constraints penalization and variables encoding. We compare three QUBO solvers: quantum annealing on quantum hardware (D-Wave Advantage), probabilistic algorithms on digital hardware and mathematical programming solvers. We analyze their QUBO resolution quality and time, and the persistence values extracted in the quantum annealing sampling process. Our results show that a linear penalization of CQKP inequality improves current best practice. Furthermore, using such a linear penalization, persistence values produced by quantum hardware in a generic way allow to match a specific CQKP metric from literature. They are therefore suitable for general purpose variable fixing in core algorithms for combinatorial optimization.
Quantum and quantum-inspired optimisation algorithms are designed to solve problems represented in binary, quadratic and unconstrained form. Combinatorial optimisation problems are therefore often formulated as Quadratic Unconstrained Binary Optimisation Problems (QUBO) to solve them with these algorithms. Moreover, these QUBO solvers are often implemented using specialised hardware to achieve enormous speedups, e.g. Fujitsu's Digital Annealer (DA) and D-Wave's Quantum Annealer. However, these are single-objective solvers, while many real-world problems feature multiple conflicting objectives. Thus, a common practice when using these QUBO solvers is to scalarise such multi-objective problems into a sequence of single-objective problems. Due to design trade-offs of these solvers, formulating each scalarisation may require more time than finding a local optimum. We present the first attempt to extend the algorithm supporting a commercial QUBO solver as a multi-objective solver that is not based on scalarisation. The proposed multi-objective DA algorithm is validated on the bi-objective Quadratic Assignment Problem. We observe that algorithm performance significantly depends on the archiving strategy adopted, and that combining DA with non-scalarisation methods to optimise multiple objectives outperforms the current scalarised version of the DA in terms of final solution quality.
Quadratic Unconstrained Binary Optimization (QUBO) problems are NP-hard problems and many real-world problems can be formulated as QUBO. Currently there are no algorithms known that can solve arbitrary instances of NP-hard problems efficiently. Therefore special-purpose hardware such as Digital Annealer, other Ising machines, as well as quantum annealers might lead to benefits in solving such problems. We study a particularly hard class of problems which can be formulated as QUBOs, namely Boolean satisfiability (SAT) problems, and specifically 3-SAT. One intriguing aspect about 3-SAT problems is that there are different transformations from 3-SAT to QUBO. We study the transformations' influence on the problem solution, using Digital Annealer as a special-purpose solver. Besides well-known transformations we investigate a novel in this context not yet discussed transformation, using less auxiliary variables and leading to very good performance. Using exact diagonalization, we explain the differences in performance originating from the different transformations. We envision that this knowledge allows for specifically engineering transformations that improve a solvers capacity to find high quality solutions. Furthermore, we show that the Digital Annealer outperforms a quantum annealer in solving hard 3-SAT instances.
Quantum annealers such as the D-Wave 2X computer are designed to natively solve Quadratic Unconstrained Binary Optimization (QUBO) problems to optimality or near optimality. Most NP-hard problems, which are hard for classical computers, can be naturally described as quadratic binary problems that contain a quadratic binary objective function and one or more constraints. Since a QUBO cannot have constraints, each such constraint has to be added to the objective function as a penalty, in order to solve on D-Wave. For a minimization problem, for instance, such penalty can be a quadratic term that gets a value zero, if the constraint is satisfied, and a large value, if it is not. In many cases, however, the penalty can significantly increase the number of quadratic terms in the resulting QUBO and make it too large to embed into the D-Wave hardware. In this paper, we develop an alternative method for formulating and embedding constraints of the type Σi=1sxi=1, which is much more scalable than the existing ones, and analyze the properties of the resulting embeddings.