S. Kobe

A recursive branch-and-bound algorithm for the exact ground state of Ising spin-glass models

Abstract A new recursive branch-and-bound algorithm is presented for solving the quadratic discrete optimization problem, which is related physically to that of finding the exact ground state of a finite Ising spin-glass model. Computational results are reported for systems with dimensions up to n = 60.

Ernst Ising—Physicist and Teacher

The Ising model is one of the standard models in statistical physics. Since 1969 more than 13800 publications using this model have appeared. In 1997 Ernst Ising celebrated his 97th birthday. Some biographical notes and milestones of the development of the Ising model are given.

Relationship between the structure of the ground level and frustration in ±J Ising lattices

Ground states of Ising lattices with ±J exchange interactions (bonds) are highly degenerate. Such degeneracy can be grouped in sets of local ensembles (LEGs) in which states are connected by single spin flips. Here we study LEGs in samples 6×6. Then we go onto the decomposition of the total ground-state degeneracy into partial degeneracies of the LEGs. Each LEG can be generated by flipping spins grouped into clusters, whose sizes are directly related to the degeneracies of the LEGs. We find that all bonds attached to a cluster are frustrated in an itinerant way. Additionally, there is rigid frustration that can be local (associated to some LEGs) or global (associated to all LEGs). The rest of the bonds constitute the diluted (unfrustrated) lattice possessing interesting properties.

Exact ground state of finite amorphous Ising systems

Abstract Branch and bound algorithms of discrete optimization programming are used to find the exact ground state of Ising systems with random antiferromagnetic exchange coupling constants.

Finding ground states of Sherrington-Kirkpatrick spin glasses with hierarchical boa and genetic algorithms

This study focuses on the problem of finding ground states of random instances of the Sherrington-Kirkpatrick (SK) spin-glass model with Gaussian couplings. While the ground states of SK spin-glass instances can be obtained with branch and bound, the computational complexity of branch and bound yields instances of not more than approximately 90 spins. We describe several approaches based on the hierarchical Bayesian optimization algorithm (hBOA) to reliably identify ground states of SK instances intractable with branch and bound, and present a broad range of empirical results on such problem instances. We argue that the proposed methodology holds a big promise for reliably solving large SK spin-glass instances to optimality with practical time complexity. The proposed approaches to identifying global optima reliably can also be applied to other problems and can be used with many other evolutionary algorithms. Performance of hBOA is compared to that of the genetic algorithm with two common crossover operators.

Low-temperature dynamics of spin glasses: walking in the energy landscape

We analyse the relationship between dynamics and configuration space structure of Ising spin glass systems. The exact knowledge of the structure of the low--energy landscape is used to study the relaxation of the system by random walk in the configuration space. The influence of the size of the valleys, clusters and energy barriers and the connectivity between them on the spin correlation function is shown.We analyse the relationship between dynamics and configuration space structure of Ising spin glass systems. The exact knowledge of the structure of the low-energy landscape is used to study the relaxation of the system by random walk in the configuration space. The influence of the size of the valleys, clusters and energy barriers and the connectivity between them on the spin correlation function is shown.

Frustration: How it can be measured

A misfit parameter is used to characterize the degree of frustration of ordered and disordered systems. It measures the increase of the ground-state energy due to frustration in comparison with that of a relevant reference state. The misfit parameter is calculated for various spin-glass models. It allows one to compare these models with each other. The extension of this concept to other combinatorial optimization problems with frustration, e.g. p-state Potts glasses, graph-partitioning problems and coloring problems is given.

An exact learning algorithm for autoassociative neural networks with binary couplings

Exact solutions for the learning problem of autoassociative networks with binary couplings are determined by a new method. The use of a branch-and-bound algorithm leads to a substantial saving of computational time compared with complete enumeration. As a result, fully connected networks with up to 40 neurons could be investigated. The network capacity is found to be close to 0.83.

Geometry of ground-state clusters in the configuration space of finite 3d ± J spin glasses

All ground states of finite disordered ± J Ising systems on cubic lattices are calculated exactly and classified into different ground-state clusters. These clusters are investigated with respect to their size and their geometric extension. We interpret the results in terms of a dependence on the fraction of antiferromagnetic couplings by a concept of free spin chains and free turnable spin groups.

Exact Energy Landscapes of Proteins Using a Coarse-Grained Model

The understanding of the relation between the amino acid sequence and the spatial structure of the protein is an open task since many decades. Some tools for the prediction of protein structures from known ones were developed, but they let unanswered fundamental questions about properties of folded proteins and the folding process itself. Recently, coarse-grained models were developed, which are able to predict protein structures with an acceptable level of accuracy using probabilistic algorithms. In this paper, we introduce a new coarse-grained model, which neglects details on the amino acid level and uses structure elements of successive amino acids as building blocks instead. Within this approximation, we use a deterministic branch and bound algorithm, which is able to find the exact ground state and the complete low-energy landscape. The agreement of the calculated ground state with the native protein structure is shown. A possible application of the model to explain experiments with membrane proteins using dynamic force microscopy is sketched.