SPIN GLASSES AND OPTIMIZATION PROBLEMS WITHOUT REPLICAS

The interest in memristors has risen due to their possible application both as memory units and as computational devices in combination with CMOS. This is in part due to their nonlinear dynamics, and a strong dependence on the circuit topology. We provide evidence that also purely memristive circuits can be employed for computational purposes. In the present paper we show that a polynomial Lyapunov function in the memory parameters exists for the case of DC controlled memristors. Such a Lyapunov function can be asymptotically approximated with binary variables, and mapped to quadratic combinatorial optimization problems. This also shows a direct parallel between memristive circuits and the Hopfield-Little model. In the case of Erdos-Renyi random circuits, we show numerically that the distribution of the matrix elements of the projectors can be roughly approximated with a Gaussian distribution, and that it scales with the inverse square root of the number of elements. This provides an approximated but direct connection with the physics of disordered system and, in particular, of mean field spin glasses. Using this and the fact that the interaction is controlled by a projector operator on the loop space of the circuit. We estimate the number of stationary points of the approximate Lyapunov function and provide a scaling formula as an upper bound in terms of the circuit topology only.

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APPLYING TABU SEARCH TO THE TWO-DIMENSIONAL ISING SPIN GLASS

International Journal of Modern Physics C ◽

10.1142/s0129183195000034 ◽

1995 ◽

Vol 06 (01) ◽

pp. 11-23 ◽

Cited By ~ 2

Author(s):

MANUEL LAGUNA ◽

PABLO LAGUNA

Keyword(s):

Tabu Search ◽

Spin Glasses ◽

Statistical Physics ◽

Optimization Problems ◽

Approximate Solutions ◽

Optimization Techniques ◽

Two Dimensional ◽

Optimal Solutions ◽

Spin Lattice ◽

Combinatorial Optimization Problems

A variety of problems in statistical physics, such as Ising-like systems, can be modeled as integer programs. Physicists have relied mostly on Monte Carlo methods to find approximate solutions to these computationally difficult problems. In some cases, optimal solutions to relatively small problems have been found using standard optimization techniques, e.g., cutting plane and branch-and-bound algorithms. Motivated by the success of tabu search (TS) in finding optimal or near-optimal solutions to combinatorial optimization problems in a number of different settings, we study the application of this methodology to Ising-like systems. Particularly, we develop a TS method to find ground states of two-dimensional spin glasses. Our method performs a search at different levels of resolution in the spin lattice, and it is designed to obtain optimal or near-optimal solutions to problem instances with several different characteristics. Results are reported for computational experiments with up to 64×64 lattices.

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FINDING MINIMA IN COMPLEX LANDSCAPES: ANNEALED, GREEDY AND RELUCTANT ALGORITHMS

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202505000765 ◽

2005 ◽

Vol 15 (09) ◽

pp. 1349-1369 ◽

Cited By ~ 1

Author(s):

PIERLUIGI CONTUCCI ◽

CRISTIAN GIARDINÀ ◽

CLAUDIO GIBERTI ◽

CECILIA VERNIA

Keyword(s):

Complex Systems ◽

Cost Function ◽

Spin Glasses ◽

Optimization Problems ◽

New Class ◽

Kirkpatrick Model ◽

Annealing Procedure ◽

The Cost ◽

Complex Landscapes

We consider optimization problems for complex systems in which the cost function has a multivalleyed landscape. We introduce a new class of dynamical algorithms which, using a suitable annealing procedure coupled with a balanced greedy-reluctant strategy drive the systems towards the deepest minimum of the cost function. Results are presented for the Sherrington–Kirkpatrick model of spin-glasses.

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Optimization problems and replica symmetry breaking in finite connectivity spin glasses

Journal of Physics A General Physics ◽

10.1088/0305-4470/31/2/012 ◽

1998 ◽

Vol 31 (2) ◽

pp. 513-529 ◽

Cited By ~ 93

Author(s):

Rémi Monasson

Keyword(s):

Symmetry Breaking ◽

Spin Glasses ◽

Optimization Problems ◽

Replica Symmetry ◽

Replica Symmetry Breaking

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Applications to Other Fields

Spin Glasses and Complexity ◽

10.23943/princeton/9780691147338.003.0007 ◽

2013 ◽

Author(s):

Daniel L. Stein ◽

Charles M. Newman

Keyword(s):

Neural Networks ◽

Combinatorial Optimization ◽

Spin Glasses ◽

Heuristic Algorithms ◽

Optimization Problems ◽

Conformational Dynamics ◽

Prebiotic Evolution ◽

Combinatorial Optimization Problems ◽

Protein Conformational Dynamics ◽

Classic Papers

This chapter explores how spin glass concepts have found use in and, in some cases, further advanced areas such as computational complexity, combinatorial optimization, neural networks, protein conformational dynamics and folding, and computer science (through the introduction of new heuristic algorithms such as simulated annealing and neural-based computation, and through new approaches to analyzing hard combinatorial optimization problems). It also introduces some “short takes” on topics that space constraints prevent covering in detail, but should be at least mentioned: prebiotic evolution, Kauffman's NK model, and the maturation of the immune response. The chapter summarizes the heart of what most people mean when they refer to spin glasses as relevant to complexity. It focuses on the early, classic papers in each subject, giving the reader a flavor of each.

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