Jump Markov chains and rejection-free Metropolis algorithms

Rosenthal, Jeffrey S.; Dote, Aki; Dabiri, Keivan; Tamura, Hirotaka; Chen, Sigeng; Sheikholeslami, Ali

doi:10.1007/s00180-021-01095-2

“…Examples include the quantum annealing machine from D-Wave Systems [16] and the coherent Ising machine (CIM) from NTT [17]. Furthermore, software simulators running on general-purpose computers such as the simulated bifurcation machine (SBM) from Toshiba [18] and dedicated but classical semiconductor chips such as the Digital Annealer from Fujitsu [19] have also been developed.…”

Section: Combinatorial Optimization Problem and The Ising Model 21 Th...mentioning

confidence: 99%

Solving Generalized Polyomino Puzzles Using the Ising Model

Takabatake

¹

,

Yanagisawa

²

,

Akiyama

³

2022

Entropy

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In the polyomino puzzle, the aim is to fill a finite space using several polyomino pieces with no overlaps or blanks. Because it is an NP-complete combinatorial optimization problem, various probabilistic and approximated approaches have been applied to find solutions. Several previous studies embedded the polyomino puzzle in a QUBO problem, where the original objective function and constraints are transformed into the Hamiltonian function of the simulated Ising model. A solution to the puzzle is obtained by searching for a ground state of Hamiltonian by simulating the dynamics of the multiple-spin system. However, previous methods could solve only tiny polyomino puzzles considering a few combinations because their Hamiltonian designs were not efficient. We propose an improved Hamiltonian design that introduces new constraints and guiding terms to weakly encourage favorable spins and pairs in the early stages of computation. The proposed model solves the pentomino puzzle represented by approximately 2000 spins with >90% probability. Additionally, we extended the method to a generalized problem where each polyomino piece could be used zero or more times and solved it with approximately 100% probability. The proposed method also appeared to be effective for the 3D polycube puzzle, which is similar to applications in fragment-based drug discovery.

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“…The Metropolis algorithm has been the most successful and influential of all the members of the Monte Carlo Method (Beichl and Sullivan, 2000). It is designed to generate a Markov chain which converges to a given target distribution π on a state space S (Rosenthal et al, 2021). With these two given elements π and S, we can generate a Markov chain by Algorithm 2.…”

Section: Background On Metropolis Algorithm For Samplingmentioning

confidence: 99%

“…The lack of the Cooling Schedule is the only difference between the Simulated Annealing algorithm and the Metropolis algorithm. Thus, like the Simulated Annealing algorithm, the Metropolis algorithm may also face the inefficiencies from the rejections (Rosenthal et al, 2021).…”

Section: Background On Metropolis Algorithm For Samplingmentioning

confidence: 99%

“…However, the Simulated Annealing algorithm may face the inefficiency of rejections. Therefore, we adopt the Rejection-Free algorithm for sampling (Rosenthal et al, 2021) into an optimization version to improve the performance of Simulated Annealing. In addition, Rejection-Free may also face inefficiency when it enters a local extreme area.…”

Section: Introductionmentioning

confidence: 99%

“…

Simulated Annealing using Metropolis steps at decreasing temperatures is widely used to solve complex combinatorial optimization problems (Kirkpatrick et al, 1983). To improve its efficiency, we can use the Rejection-Free version of the Metropolis algorithm which avoids the inefficiency of rejections by considering all the neighbors at each step (Rosenthal et al, 2021). To avoid the algorithm getting stuck in a local extreme area, we propose an enhanced version of Rejection-Free called Partial Neighbor Search, which only considers random part of the neighbors while applying the Rejection-Free technique.

…”

mentioning

confidence: 99%

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Optimization via Rejection-Free Partial Neighbor Search

Chen¹,

Rosenthal²,

Dote³

et al. 2022

Preprint

Self Cite

0

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Simulated Annealing using Metropolis steps at decreasing temperatures is widely used to solve complex combinatorial optimization problems (Kirkpatrick et al, 1983). To improve its efficiency, we can use the Rejection-Free version of the Metropolis algorithm which avoids the inefficiency of rejections by considering all the neighbors at each step (Rosenthal et al, 2021). To avoid the algorithm getting stuck in a local extreme area, we propose an enhanced version of Rejection-Free called Partial Neighbor Search, which only considers random part of the neighbors while applying the Rejection-Free technique. In this paper, we apply these methods to several examples such as quadratic unconstrained binary optimization (QUBO) problems to demonstrate superior performance of the Rejection-Free Partial Neighbor Search algorithm.

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