scholarly journals Ising Model Optimization Problems on a FPGA Accelerated Restricted Boltzmann Machine

2020 ◽  
Author(s):  
Saavan Patel ◽  
Lili Chen ◽  
Philip Canoza ◽  
Sayeef Salahuddin
Keyword(s):  
Optimization Problems ◽  
Restricted Boltzmann Machine ◽  
Model Optimization ◽  
Boltzmann Machine ◽  
Stochastic Sampling ◽  
Combinatorial Optimization Problems ◽  
Field Programmable ◽  
Max Cut Problem ◽  
Asymptotic Scaling ◽  
Better Than

Abstract In this work we demonstrate usage of the Restricted Boltzmann Machine (RBM) as a stochastic neural network capable of solving NP-Hard Combinatorial Optimization problems efficiently. By mapping the RBM onto a reconfigurable Field Programmable Gate Array (FPGA), we can effectively hardware accelerate the RBM's stochastic sampling algorithm. We benchmark the RBM against the DWave 2000Q Quantum Adiabatic Computer and the Optical Coherent Ising Machine on two such optimization problems: the MAX-CUT problem and the Sherrington-Kirkpatrick (SK) spin glass. The hardware accelerated RBM shows asymptotic scaling either similar or better than these other accelerators. This leads to 107x and 105x time to solution improvement compared to the DWave 2000Q on the MAX-CUT and SK problems respectively, along with a 150x and 1000x improvement compared to the Coherent Ising Machine annealer on those problems. By utilizing commodity hardware running at room temperature, the RBM shows potential for immediate and scalable use.

scholarly journals Combinatorial optimization by simulating adiabatic bifurcations in nonlinear Hamiltonian systems

2019 ◽  
Vol 5 (4) ◽  
pp. eaav2372 ◽  
Author(s):  
Hayato Goto ◽  
Kosuke Tatsumura ◽  
Alexander R. Dixon
Keyword(s):  
Combinatorial Optimization ◽  
Hamiltonian Systems ◽  
Large Scale ◽  
Optimization Problems ◽  
Approximate Solutions ◽  
Combinatorial Optimization Problems ◽  
Bifurcation Phenomena ◽  
Field Programmable ◽  
Max Cut Problem ◽  
Adiabatic Optimization

Combinatorial optimization problems are ubiquitous but difficult to solve. Hardware devices for these problems have recently been developed by various approaches, including quantum computers. Inspired by recently proposed quantum adiabatic optimization using a nonlinear oscillator network, we propose a new optimization algorithm simulating adiabatic evolutions of classical nonlinear Hamiltonian systems exhibiting bifurcation phenomena, which we call simulated bifurcation (SB). SB is based on adiabatic and chaotic (ergodic) evolutions of nonlinear Hamiltonian systems. SB is also suitable for parallel computing because of its simultaneous updating. Implementing SB with a field-programmable gate array, we demonstrate that the SB machine can obtain good approximate solutions of an all-to-all connected 2000-node MAX-CUT problem in 0.5 ms, which is about 10 times faster than a state-of-the-art laser-based machine called a coherent Ising machine. SB will accelerate large-scale combinatorial optimization harnessing digital computer technologies and also offer a new application of computational and mathematical physics.

scholarly journals Restricted Boltzmann Machine-Assisted Estimation of Distribution Algorithm for Complex Problems

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-13 ◽  
Author(s):  
Lin Bao ◽  
Xiaoyan Sun ◽  
Yang Chen ◽  
Guangyi Man ◽  
Hui Shao
Keyword(s):  
Optimization Problems ◽  
Probability Model ◽  
Computational Cost ◽  
Evolutionary Process ◽  
Estimation Of Distribution Algorithm ◽  
Restricted Boltzmann Machine ◽  
Model Management ◽  
Boltzmann Machine ◽  
Complex Problems ◽  
Estimation Of Distribution

A novel algorithm, called restricted Boltzmann machine-assisted estimation of distribution algorithm, is proposed for solving computationally expensive optimization problems with discrete variables. First, the individuals are evaluated using expensive fitness functions of the complex problems, and some dominant solutions are selected to construct the surrogate model. The restricted Boltzmann machine (RBM) is built and trained with the dominant solutions to implicitly extract the distributed representative information of the decision variables in the promising subset. The visible layer’s probability of the RBM is designed as the sampling probability model of the estimation of distribution algorithm (EDA) and is updated dynamically along with the update of the dominant subsets. Second, according to the energy function of the RBM, a fitness surrogate is developed to approximate the expensive individual fitness evaluations and participates in the evolutionary process to reduce the computational cost. Finally, model management is developed to train and update the RBM model with newly dominant solutions. A comparison of the proposed algorithm with several state-of-the-art surrogate-assisted evolutionary algorithms demonstrates that the proposed algorithm effectively and efficiently solves complex optimization problems with smaller computational cost.

Bio-Inspired Algorithm for the Shortest Path According to the Maximum Time for Each Trial

2013 ◽  
Vol 717 ◽  
pp. 455-459
Author(s):  
Seung Gwan Lee ◽  
Seung Won Lee
Keyword(s):  
Simulation Experiment ◽  
State Transition ◽  
Optimization Problems ◽  
Ant Colony ◽  
Transition Rule ◽  
Ant Colony System ◽  
Combinatorial Optimization Problems ◽  
Maximum Time ◽  
Acs Algorithm ◽  
Better Than

Ant Colony System (ACS) is a new meta heuristics algorithms to solve hard combinatorial optimization problems. In this paper, we propose hybrid ant colony algotirhm that is searching the second best edge first in the state transition rule and updating the pheromone on edges applying the visited number of edge in the globally best tour. And we evaluate the proposed algorithm according to the maximum time for each trial. The results of a simulation experiment demonstrate that the proposed algorithm is better than, or, at least as good as, that of ACS algorithm in the most sets.

scholarly journals Reconfigurable Stochastic neurons based on tin oxide/MoS2 hetero-memristors for simulated annealing and the Boltzmann machine

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Xiaodong Yan ◽  
Jiahui Ma ◽  
Tong Wu ◽  
Aoyang Zhang ◽  
Jiangbin Wu ◽  
...  
Keyword(s):  
Simulated Annealing ◽  
Combinatorial Optimization ◽  
Tin Oxide ◽  
Optimization Problems ◽  
Process Efficiency ◽  
Np Hard ◽  
Boltzmann Machine ◽  
Combinatorial Optimization Problems ◽  
Cooling Strategies ◽  
Neuromorphic Hardware

AbstractNeuromorphic hardware implementation of Boltzmann Machine using a network of stochastic neurons can allow non-deterministic polynomial-time (NP) hard combinatorial optimization problems to be efficiently solved. Efficient implementation of such Boltzmann Machine with simulated annealing desires the statistical parameters of the stochastic neurons to be dynamically tunable, however, there has been limited research on stochastic semiconductor devices with controllable statistical distributions. Here, we demonstrate a reconfigurable tin oxide (SnOx)/molybdenum disulfide (MoS2) heterogeneous memristive device that can realize tunable stochastic dynamics in its output sampling characteristics. The device can sample exponential-class sigmoidal distributions analogous to the Fermi-Dirac distribution of physical systems with quantitatively defined tunable “temperature” effect. A BM composed of these tunable stochastic neuron devices, which can enable simulated annealing with designed “cooling” strategies, is conducted to solve the MAX-SAT, a representative in NP-hard combinatorial optimization problems. Quantitative insights into the effect of different “cooling” strategies on improving the BM optimization process efficiency are also provided.

scholarly journals Clustering Improves the Goemans–Williamson Approximation for the Max-Cut Problem

Computation ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 75
Author(s):  
Angel E. Rodriguez-Fernandez ◽  
Bernardo Gonzalez-Torres ◽  
Ricardo Menchaca-Mendez ◽  
Peter F. Stadler
Keyword(s):  
Lower Bounds ◽  
Clustering Algorithm ◽  
Optimization Problems ◽  
Clustering Algorithms ◽  
Quadratic Program ◽  
Combinatorial Optimization Problems ◽  
Integer Spin ◽  
Integer Quadratic Programming ◽  
Max Cut Problem ◽  
Global Optima

MAX-CUT is one of the well-studied NP-hard combinatorial optimization problems. It can be formulated as an Integer Quadratic Programming problem and admits a simple relaxation obtained by replacing the integer “spin” variables xi by unitary vectors v→i. The Goemans–Williamson rounding algorithm assigns the solution vectors of the relaxed quadratic program to a corresponding integer spin depending on the sign of the scalar product v→i·r→ with a random vector r→. Here, we investigate whether better graph cuts can be obtained by instead using a more sophisticated clustering algorithm. We answer this question affirmatively. Different initializations of k-means and k-medoids clustering produce better cuts for the graph instances of the most well known benchmark for MAX-CUT. In particular, we found a strong correlation of cluster quality and cut weights during the evolution of the clustering algorithms. Finally, since in general the maximal cut weight of a graph is not known beforehand, we derived instance-specific lower bounds for the approximation ratio, which give information of how close a solution is to the global optima for a particular instance. For the graphs in our benchmark, the instance specific lower bounds significantly exceed the Goemans–Williamson guarantee.

A CONTINUATION APPROACH USING NCP FUNCTION FOR SOLVING MAX-CUT PROBLEM

2009 ◽  
Vol 26 (04) ◽  
pp. 445-456
Author(s):  
FENGMIN XU ◽  
CHENGXIAN XU ◽  
JIUQUAN REN
Keyword(s):  
Optimization Problem ◽  
Optimization Problems ◽  
Continuous Optimization ◽  
Test Problems ◽  
Feasible Direction ◽  
Combinatorial Optimization Problems ◽  
Ncp Function ◽  
Continuous Optimization Problem ◽  
Feasible Direction Method ◽  
Max Cut Problem

A continuous approach using NCP function for approximating the solution of the max-cut problem is proposed. The max-cut problem is relaxed into an equivalent nonlinearly constrained continuous optimization problem and a feasible direction method without line searches is presented for generating an optimal solution of the relaxed continuous optimization problem. The convergence of the algorithm is proved. Numerical experiments and comparisons on some max-cut test problems show that we can get the satisfactory solution of max-cut problems with less computation time. Furthermore, this is the first time that the feasible direction method is combined with NCP function for solving max-cut problem, and similar idea can be generalized to other combinatorial optimization problems.

scholarly journals Assessment of image generation by quantum annealer

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Takehito Sato ◽  
Masayuki Ohzeki ◽  
Kazuyuki Tanaka
Keyword(s):  
Machine Learning ◽  
Optimization Problems ◽  
A Priori ◽  
Monte Carlo Sampling ◽  
Boltzmann Distribution ◽  
Generative Model ◽  
Production Model ◽  
Boltzmann Machine ◽  
Quantum Annealing ◽  
Combinatorial Optimization Problems

AbstractQuantum annealing was originally proposed as an approach for solving combinatorial optimization problems using quantum effects. D-Wave Systems has released a production model of quantum annealing hardware. However, the inherent noise and various environmental factors in the hardware hamper the determination of optimal solutions. In addition, the freezing effect in regions with weak quantum fluctuations generates outputs approximately following a Gibbs–Boltzmann distribution at an extremely low temperature. Thus, a quantum annealer may also serve as a fast sampler for the Ising spin-glass problem, and several studies have investigated Boltzmann machine learning using a quantum annealer. Previous developments have focused on comparing the performance in the standard distance of the resulting distributions between conventional methods in classical computers and sampling by a quantum annealer. In this study, we focused on the performance of a quantum annealer as a generative model from a different aspect. To evaluate its performance, we prepared a discriminator given by a neural network trained on an a priori dataset. The evaluation results show a higher performance of quantum annealer compared with the classical approach for Boltzmann machine learning in training of the generative model. However the generation of the data suffers from the remanent quantum fluctuation in the quantum annealer. The quality of the generated images from the quantum annealer gets worse than the ideal case of the quantum annealing and the classical Monte-Carlo sampling.

scholarly journals A MULTI-COLONY ANT SYSTEM FOR COMBINATORIAL OPTIMIZATION PROBLEM

2012 ◽  
Vol 11 (02) ◽  
pp. 1250012
Author(s):  
RONG-LONG WANG ◽  
XIAO-FAN ZHOU ◽  
LI-QING ZHAO ◽  
ZE-WEI XIA
Keyword(s):  
Combinatorial Optimization ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Natural World ◽  
Search Procedure ◽  
Ant System ◽  
Combinatorial Optimization Problems ◽  
Time Period ◽  
The Traveling Salesman Problem ◽  
Better Than

A multi-colony ant system (MAS) is proposed for the combinatorial optimization problems. The proposed MAS is inspired by the knowledge that there are many colonies of ants in the natural world and organized with multiple colonies of ants. At first, ants perform solution search procedure by cooperating with each other in the same colony until no better solution is found after a certain time period. Then, communication between different colonies is performed to build new pheromone distributions for each colony, and ants start their search procedure again in each separate colony, based on the new pheromone distribution. The proposed algorithm is tested by simulating the traveling salesman problem (TSP). Simulation results show that the proposed method performs better than the traditional ACO.

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