Quantum computing based hybrid solution strategies for large-scale discrete-continuous optimization problems

“Heuristic techniques such as simulated annealing [6], genetic algorithms [7], tabu search [8] and others have also grown popular owing to their easy implementation and little prior knowledge requirement of the optimization problem.”

“QC’s major applications can be perceived in areas of optimization, machine learning, cryptography, and quantum chemistry [16].”

“This is evident from QC techniques like Shor’s algorithm for integer factorization [11], Grover’s search algorithm for unstructured databases [12], quantum algorithm for linear system of equations [13], and many more [14].”

“This is evident from QC techniques like Shor’s algorithm for integer factorization [11], Grover’s search algorithm for unstructured databases [12], quantum algorithm for linear system of equations [13], and many more [14].”

“The proposed hybrid QC-MIQP stepwise decomposition method is based on Bender’s decomposition [65], by considering the above manufacturing cell formation problem given in Eq.”

“This is evident from QC techniques like Shor’s algorithm for integer factorization [11], Grover’s search algorithm for unstructured databases [12], quantum algorithm for linear system of equations [13], and many more [14].”

“A Hamiltonian is a mathematical description of the physical system in terms of its energies and corresponds to the objective function of an optimization problem [29].”

“Quantum annealing is a quantum analogue of classical simulated annealing that permits quantum tunneling as shown in Figure 1c to aid in exploring low-cost solutions and ultimately yield a global minimum [25].”

“Heuristic and metaheuristic algorithms can be directly applied to several variants of the vehicle routing problem [41, 72], but such techniques are context dependent and require careful parameter tuning to obtain good-quality solutions [73, 74].”

“Due to the versatile nature and richness in terms of real-world applicability, the vehicle routing problem has attracted significant attention [69].”

“In addition, the vehicle routing problem leads to challenging formulations that belong to the NP-hard computational complexity class and requires development of sophisticated solution strategies [70].”

“All the integer programming formulations of molecular conformation problem are modeled with Pyomo [55] and solved with MILP solver Gurobi.”

“This is evident from QC techniques like Shor’s algorithm for integer factorization [11], Grover’s search algorithm for unstructured databases [12], quantum algorithm for linear system of equations [13], and many more [14].”

“Job-shop scheduling problems belong to the class of most intractable NP-hard combinatorial optimization problems, and pose a significant computational challenge due to its large and complex search space [43].”

“Heuristic and metaheuristic algorithms can be directly applied to several variants of the vehicle routing problem [41, 72], but such techniques are context dependent and require careful parameter tuning to obtain good-quality solutions [73, 74].”

“This computational challenge could be handled by advancements in computers with high processing speeds, but saturation limits of Moore’s law render the possibility of rising processing speeds unlikely in the coming years [9].”

“Over the past years, deterministic optimization techniques, especially nonlinear programming and mixed-integer nonlinear programming (MINLP) algorithms, have received growing attention from academia and industry [3, 4].”

“Several heuristic solution techniques have also been proposed to solve complex job-shop scheduling problems, but they are unable to find high-quality solutions if large-size and complex search spaces are involved [56, 57].”

“Heuristic and metaheuristic algorithms can be directly applied to several variants of the vehicle routing problem [41, 72], but such techniques are context dependent and require careful parameter tuning to obtain good-quality solutions [73, 74].”

“Despite the algorithmic and applications-oriented advances in global optimization [5], large-scale nonconvex MINLP problems could still be computationally very expensive to solve with the state-of-the-art deterministic global optimization algorithms.”

“The branch-and-cut method and generalized Bender’s decomposition based methods are some of them [63, 64].”

“The proposed models and solution algorithm draw inspiration from the hybrid algorithm proposed by Jain and Grossmann [59].”

“The probability of recovering the global optimal solution is highly dependent on the embedding and annealing schedule [37, 38].”

“Four applications are considered in this paper, namely the molecular conformation problem [39], the job-shop scheduling problem [40], the manufacturing cell formation problem and the vehicle routing problem [41].”

“While the problem of finding an optimal graph-minor is itself NP-hard, an efficient embedding for many graphs can be found systematically with heuristic techniques [34].”

“This problem is concerned with determining an optimal set of routes for a fleet of vehicles in order to serve a given set of customers or locations, in order to minimize the total transportation cost [68].”

“Although it has been shown that some optimization problems relevant to energy systems can be solved using quantum computers, their performance deteriorates with increasing size and complexity [17].”

“The first step of cellular manufacturing system design is cell formation which involves selecting parts and machines that will be allocated to each cell [60].”

“Decomposition-based algorithms and hybrid algorithms that combine exact solution methods with constraint programming reduce the combinatorial complexity of job-shop scheduling problems, and have proven effective for several real-world industrial-scale problems [58, 59].”

“For example, QC-based solution algorithms for operational planning problems can only solve small-scale instances and do not produce results that are comparable with those obtained from the state-of-the-art classical computing approaches [20, 21].”

“The probability of recovering the global optimal solution is highly dependent on the embedding and annealing schedule [37, 38].”

“Heuristic and metaheuristic algorithms can be directly applied to several variants of the vehicle routing problem [41, 72], but such techniques are context dependent and require careful parameter tuning to obtain good-quality solutions [73, 74].”

“Over the past years, deterministic optimization techniques, especially nonlinear programming and mixed-integer nonlinear programming (MINLP) algorithms, have received growing attention from academia and industry [3, 4].”

“Minimization of total potential energy associated with configuration of atoms in a molecule is known as the molecular conformation problem [45].”

“They are the gate-model quantum computer and the annealing-based quantum computer [22, 23].”

“Such applications require solving the molecular conformation problem to global optimality [46].”

“Since the Chimera lattice is not fully connected, mapping of variables to physical qubits uses the process of minor-embedding and is crucial to problem solving [30-33].”

“Since the Chimera lattice is not fully connected, mapping of variables to physical qubits uses the process of minor-embedding and is crucial to problem solving [30-33].”

“The branch-and-cut method and generalized Bender’s decomposition based methods are some of them [63, 64].”

“Sophisticated exact solution techniques based on integer programming like branch-and-cut and branch-and-prize can solve medium-sized vehicle routing problems, but success of these methods depend on the model formulation [71].”

“Heuristic solution techniques for the molecular conformation problem are computationally expensive, depend heavily on carefully chosen parameters, and do not guarantee a global optimum [50].”

“The discretized molecular conformation problem can be formulated as a quadratic assignment problem [47].”

“The manufacturing cell formation problems are solved using the proposed hybrid QC-MIQP stepwise decomposition method that is based on generalized Bender’s decomposition [65], which has been used in batch manufacturing systems [66, 67].”

“of-the-art classical computing approaches [20, 21].”

“Several heuristic solution techniques have also been proposed to solve complex job-shop scheduling problems, but they are unable to find high-quality solutions if large-size and complex search spaces are involved [56, 57].”

“The manufacturing cell formation problems are solved using the proposed hybrid QC-MIQP stepwise decomposition method that is based on generalized Bender’s decomposition [65], which has been used in batch manufacturing systems [66, 67].”

“Exact techniques to solve the molecular conformation problem have also been proposed, but they could perform poorly as the size of the molecule increases [45, 51].”

“Since the Chimera lattice is not fully connected, mapping of variables to physical qubits uses the process of minor-embedding and is crucial to problem solving [30-33].”

“They are the gate-model quantum computer and the annealing-based quantum computer [22, 23].”

“10 predict the native structure of sequence of molecules, and it can be considered as a simplified form of predicting native structures of residues in case of protein folding [39, 47-49].”

“This approach has been successful in finding the minimum energy conformations for very large molecules [52, 53].”

“The molecular conformation problem is a building block of molecular design, having major implications in the field of drug design and product design [42].”

“Such fractional objective function formulations have been previously applied to real-world large-scale routing problems with inventory management [77].”

“Hybridization of exact and metaheuristic algorithms has proven to be a more viable option that uses complementary strengths of both techniques, and overcomes difficulties associated with determining mutually separable cells [62].”

“From an optimization perspective, annealingbased quantum computers are much closer to discrete optimization problems than the gate-model quantum computers [27].”

“Therefore, harnessing the complementary strengths of classical and quantum computers to solve complex large-scale optimization problems has become the main strategy for near-term and mid-term solution [18, 19].”

“Mapping the objective function onto the physical quantum processing unit is followed by the realization of quantum annealing process [35] which searches for low-energy solutions of the corresponding problem Hamiltonian [36].”

“Since the Chimera lattice is not fully connected, mapping of variables to physical qubits uses the process of minor-embedding and is crucial to problem solving [30-33].”

“Mathematical programming formulations for the vehicle routing problems can be broadly classified into two categories, namely the vehicle flow formulation and the set partitioning formulation [75].”

“Therefore, harnessing the complementary strengths of classical and quantum computers to solve complex large-scale optimization problems has become the main strategy for near-term and mid-term solution [18, 19].”

“10 predict the native structure of sequence of molecules, and it can be considered as a simplified form of predicting native structures of residues in case of protein folding [39, 47-49].”