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Quantum devices can be used to solve constrained combinatorial optimization (COPT) problems thanks to the use of penalization methods to embed the COPT problem’s constraints in its objective to obtain a quadratic unconstrained binary optimization (QUBO) reformulation of the COPT. However, the particular way in which this penalization is carried out...
Citations
... Indeed, QAOA derives from the Quantum Annealing algorithm of Kadowaki and Nishimori (1998), algorithm that takes as input an Ising Model, a model of ferromagnetism in statistical mechanics, which is equivalent to a QUBO problem. Thus, natively, the comparison of different formulations of QUBO models for a given problem recently raised interest in the community, for instance for the graph coloring problem (Tabi et al., 2020) or for Max-k-colorable subgraph problem (Quintero et al., 2022). However, there are in fact no inherent degree limitations of the objective function. ...
Quantum Approximate Optimization Algorithm (QAOA) is one of the most short-term promising quantum-classical algorithm to solve unconstrained combinatorial optimization problems. It alternates between the execution of a parametrized quantum circuit and a classical optimization. There are numerous levers for enhancing QAOA performances, such as the choice of quantum circuit meta-parameters or the choice of the classical optimizer. In this paper, we stress on the importance of the input problem formulation by illustrating it with the resolution of an industrial railway timetabling problem. Specifically, we present a generic method to reformulate any polynomial problem into a Polynomial Unconstrained Binary Optimization (PUBO) problem, with a specific formulation imposing penalty terms to take binary values when the constraints are linear. We also provide a generic reformulation into a Quadratic Unconstrained Binary Optimization (QUBO) problem. We then conduct a numerical comparison between the PUBO with binary penalty terms and the QUBO formulations proposed on a railway timetabling problem solved with QAOA. Our results illustrate that the PUBO reformulation outperforms the QUBO one for the problem at hand.
... This proliferation has reignited interest in reformulation techniques tailored to this type of problems [see, e.g. [12][13][14][15][16][17]. ...
... 21] and gate-based [see, e.g. 22] quantum devices can address the solution of QUBO problems, and potentially have quantum supremacy [see, e.g., 23] over classical computers on this task, it is clearly of interest to study whether or how combinatorial optimization problems that do not have a natural QUBO formulation (e.g., the stable set problem) can be reformulated as a QUBO [see, e.g., [12][13][14][15]24] A classical approach to solving optimization problems with complex or computationally intensive constraints is the use of Lagrangian relaxations [see, e.g., 20,25]. By relaxing (or dualizing) the constraints and solving the resulting simpler problem iteratively, Lagrangian relaxation provides a systematic approach to finding near-optimal solutions efficiently. ...
... On the one hand, general Lagrangian duality results for nonconvex optimization [see, e.g., 34-38, among many others] often lack constructive dual attainment characterizations necessary for deriving the practical reformulations that are of interest here. On the other hand, Lagrangian reformulation results exist for specific optimization problem classes, such as linearly constrained pure binary quadratic optimization [see, e.g., [12][13][14][15][16][17], mixed-integer linear optimization [39], and mixed-integer convex quadratic optimization [40,41]. By deriving constructive strong duality and dual attainment results, we provide Lagrangian reformulation results with practical applications for a broad class of nonconvex optimization problems, including the latter problems metioned. ...
In recent years, there has been a surge of interest in studying different ways to reformulate nonconvex optimization problems, especially those that involve binary variables. This interest surge is due to advancements in computing technologies, such as quantum and Ising devices, as well as improvements in quantum and classical optimization solvers that take advantage of particular formulations of nonconvex problems to tackle their solutions. Our research characterizes the equivalence between equality-constrained nonconvex optimization problems and their Lagrangian relaxation, enabling the aforementioned new technologies to solve these problems. In addition to filling a crucial gap in the literature, our results are readily applicable to many important situations in practice. To obtain these results, we bridge between specific optimization problem characteristics and broader, classical results on Lagrangian duality for general nonconvex problems. Further, our approach takes a comprehensive approach to the question of equivalence between problem formulations. We consider this question not only from the perspective of the problem's objective but also from the viewpoint of its solution. This perspective, often overlooked in existing literature, is particularly relevant for problems featuring continuous and binary variables.
... However, it is important to prepare the ground for the time when bigger, more resilient to noise quantum computers will become available, by exploring which formulations of combinatorial optimization problems are more suitable for each task. This is a very active area of research, and for instance, in the last few years, several authors have studied the merits of different quantum formulations of the Travelling Salesperson Problem and its variants [23][24][25], of graph-coloring problems [26,27], of workflow scheduling problems [28,29], and of vehicle routing problems [30,31]. ...
... In practice, however, choosing the right formulation can be of paramount importance in practical applications of quantum optimization techniques. In recent years, several researchers have explored the performance of quantum optimization techniques under different formulations for famous problems such as the Travelling Salesperson Problem [23][24][25], graph coloring [26,27], vehicle routing [30,31], and workflow scheduling [28,29], showing that there can be significant differences among them. ...
The aim of a ranking aggregation problem is to combine several rankings into a single one that best represents them. A common method for solving this problem is due to Kemeny and selects as the aggregated ranking the one that minimizes the sum of the Kendall distances to the rankings to be aggregated. Unfortunately, the identification of the said ranking—called the Kemeny ranking—is known to be a computationally expensive task. In this paper, we study different ways of computing the Kemeny ranking with quantum optimization algorithms, and in particular, we provide some alternative formulations for the search for the Kemeny ranking as an optimization problem. To the best of our knowledge, this is the first time that this problem is addressed with quantum techniques. We propose four different ways of formulating the problem, one novel to this work. Two different quantum optimization algorithms—Quantum Approximate Optimization Algorithm and Quantum Adiabatic Computing—are used to evaluate each of the different formulations. The experimental results show that the choice of the formulation plays a big role on the performance of the quantum optimization algorithms.
... Therefore, many optimization problems can be formulated as Ising models to find the ground state, or the lowest energy configuration. So that, solving the Ising model becomes a general method for solving many NP problems, like partitioning problems [2], linear programming [1], [3], [5], inequality problems [6], coloring problems [2], [7] and so on. However, the Ising model is known to be NP-hard (Non-deterministic Polynomial Hard) problem [8]. ...
The ground state search of the Ising model can be used to solve many combinatorial optimization problems. Under the current computer architecture, an Ising ground state search algorithm suitable for hardware computing is necessary for solving practical problems. Inspired by the potential energy conversion of springs, we propose a point convolutional neural network algorithm for ground state search based on spring vibration model, called Spring-Ising Algorithm. Spring-Ising Algorithm regards the spin as a moving mass point connected to a spring and establish the equation of motion for all spins. Spring-Ising Algorithm can be mapped on the GPU or AI chips through the basic structure of the neural network for fast and efficient parallel computing. The algorithm has very productive results for solving the Ising model and has been test in the recognized test benchmark K2000. The algorithm introduces the concept of dynamic equilibrium to achieve a more detailed local search by dynamically adjusting the weight of the Ising model in the spring oscillation model. Finally, there is the simple hardware test speed evaluation. Spring-Ising Algorithm can provide the possibility to calculate the Ising model on a chip which focuses on accelerating neural network calculations.
... This aligns with the interpretation that the penalty weight represents a trade-off between satisfying the constraints versus optimizing the objective. These empirical results also corroborate the analytical results of [62], which state that the minimum valid penalty weight for the stable set of a graph is 1. Given that maximum clique represents the maximum clique problem as finding the stable set of the graph built with the complement of the original edges, the bound on the penalty weight is valid. ...
Recent years have seen significant advances in quantum/quantum-inspired technologies capable of approximately searching for the ground state of Ising spin Hamiltonians. The promise of leveraging such technologies to accelerate the solution of difficult optimization problems has spurred an increased interest in exploring methods to integrate Ising problems as part of their solution process, with existing approaches ranging from direct transcription to hybrid quantum-classical approaches rooted in existing optimization algorithms. Due to the heuristic and black-box nature of the underlying Ising solvers, many such approaches have limited optimality guarantees. While some hybrid algorithms may converge to global optima, their underlying classical algorithms typically rely on exhaustive search, making it unclear if such algorithmic scaffolds are primed to take advantage of speed-ups that the Ising solver may offer. In this paper, we propose a framework for solving mixed-binary quadratic programs (MBQP) to global optimality using black-box and heuristic Ising solvers. We show the exactness of a convex copositive reformulation of MBQPs, which we propose to solve via a hybrid quantum-classical cutting-plane algorithm. The classical portion of this hybrid framework is guaranteed to be polynomial time, suggesting that when applied to NP-hard problems, the complexity of the solution is shifted onto the subroutine handled by the Ising solver.
... Şeker et al. (2020) provided computational comparisons for the QUBO formulations of the following problems on different solvers: (i) quadratic assignment, (ii) quadratic cycle partition, and (iii) selective graph coloring. Finally, Quintero et al. (2021) proposed a QUBO formulation for the maximum k-colorable subgraph problem. ...
... for every partition j ∈ P do 6: One can always set a "big" w in formulations (19) and (20) to ensure that an optimal solution of the unconstrained formulations always represents a feasible solution for the max k-cut problem. However, large values of the elements of w have a detrimental effect on the performance of quantum computing algorithms (Quintero et al., 2021). Theorems 3 and 4 propose smallest values for elements of the penalty vector w. ...
Recent claims on "solving'' combinatorial optimization problems via quantum computers have attracted many researchers to work on quantum algorithms. The max k-cut problem is a challenging combinatorial optimization problem with multiple notorious mixed integer linear optimization (MILO) formulations. Motivated by the recent progress of classical solvers in handling quadratic optimization problems, we revisit the binary quadratic optimization (BQO) formulation of Carlson and Nemhauser (Operations Research, 1966) and provide theoretical and computational comparisons between different mixed integer optimization formulations of the max k-cut problem. While no claim on "quantum advantage'' is provided, we propose quadratic unconstrained binary optimization (QUBO) formulations with tight penalty coefficients and design quantum circuits for finding a feasible solution of the problem by a quantum approximate optimization algorithm (QAOA). To accelerate the solving process and handle large-scale instances of the problem, we propose a preprocessing procedure employed in classical and quantum contexts. (i) peeling and folding operations as well as biconnected decomposition, and (ii) convexification procedures for the BQO model. Finally, we test the performance of our classical and quantum-inspired formulations on an extensive set of instances. Codes and data are available on GitHub.
The ground state search of the Ising model can be used to solve many combinatorial optimization problems. Under the current computer architecture, an Ising ground state search algorithm suitable for hardware computing is necessary for solving practical problems. Inspired by the potential energy conversion of the springs, we propose the Spring-Ising Algorithm, a point convolutional neural network algorithm for ground state search based on the spring vibration model. Spring-Ising Algorithm regards the spin as a moving mass point connected to a spring and establishes the equation of motion for all spins. Spring-Ising Algorithm can be mapped on AI chips through the basic structure of the neural network for fast and efficient parallel computing. The algorithm has shown promising results in solving the Ising model and has been tested in the recognized test benchmark K2000. The optimal results of this algorithm after 10,000 steps of iteration are 2.9% of all results. The algorithm introduces the concept of dynamic equilibrium to achieve a more detailed local search by dynamically adjusting the weight of the Ising model in the spring oscillation model. Spring-Ising Algorithm offers the possibility to calculate the Ising model on a chip which focuses on accelerating neural network calculations.
The application of quantum computing to combinatorial optimization problems is attracting increasing research interest, resulting in diverse approaches and research streams. This study aims at identifying, classifying, and understanding existing solution approaches as well as typical use cases in the field. The obtained classification schemes are based on a full-text analysis of 156 included papers. Our results can be used by researchers and practitioners to (i) better understand adaptations to and utilizations of existing gate-based and quantum annealing approaches, and (ii) identify typical use cases for quantum computing in areas like graph optimization, routing and scheduling.
