Dynamical Analysis of Continuous Higher-Order Hopfield Networks for Combinatorial Optimization

2005 ◽  
Vol 17 (8) ◽  
pp. 1802-1819 ◽  
Author(s):  
Miguel Atencia ◽  
Gonzalo Joya ◽  
Francisco Sandoval
Keyword(s):  
Combinatorial Optimization ◽  
Fixed Points ◽  
Optimization Problems ◽  
Target Function ◽  
Higher Order ◽  
Dynamical Analysis ◽  
Hopfield Networks ◽  
Combinatorial Optimization Problems ◽  
Continuous Equation ◽  
The Stability

In this letter, the ability of higher-order Hopfield networks to solve combinatorial optimization problems is assessed by means of a rigorous analysis of their properties. The stability of the continuous network is almost completely clarified: (1) hyperbolic interior equilibria, which are unfeasible, are unstable; (2) the state cannot escape from the unitary hypercube; and (3) a Lyapunov function exists. Numerical methods used to implement the continuous equation on a computer should be designed with the aim of preserving these favorable properties. The case of nonhyperbolic fixed points, which occur when the Hessian of the target function is the null matrix, requires further study. We prove that these nonhyperbolic interior fixed points are unstable in networks with three neurons and order two. The conjecture that interior equilibria are unstable in the general case is left open.

Artificial Higher Order Neural Networks for Modeling Combinatorial Optimization Problems

2013 ◽  
pp. 44-57
Author(s):  
Yuxin Ding
Keyword(s):  
Neural Networks ◽  
Combinatorial Optimization ◽  
Convergence Rates ◽  
Optimization Problems ◽  
High Order ◽  
Hopfield Network ◽  
Hopfield Networks ◽  
Combinatorial Optimization Problems ◽  
Complete Problems ◽  
Higher Order Neural Networks

Traditional Hopfield networking has been widely used to solve combinatorial optimization problems. However, high order Hopfiled networks, as an expansion of traditional Hopfield networks, are seldom used to solve combinatorial optimization problems. In theory, compared with low order networks, high order networks have better properties, such as stronger approximations and faster convergence rates. In this chapter, the authors focus on how to use high order networks to model combinatorial optimization problems. Firstly, the high order discrete Hopfield Network is introduced, then the authors discuss how to find the high order inputs of a neuron. Finally, the construction method of energy function and the neural computing algorithm are presented. In this chapter, the N queens problem and the crossbar switch problem, which are NP-complete problems, are used as examples to illustrate how to model practical problems using high order neural networks. The authors also discuss the performance of high order networks for modeling the two combinatorial optimization problems.

Semidefinite optimization

Acta Numerica ◽  
2001 ◽  
Vol 10 ◽  
pp. 515-560 ◽  
Author(s):  
M. J. Todd
Keyword(s):  
Combinatorial Optimization ◽  
Differential Inclusion ◽  
Computational Methods ◽  
Duality Theory ◽  
Optimization Problems ◽  
Symmetric Matrix ◽  
Positive Semidefinite ◽  
Semidefinite Optimization ◽  
Combinatorial Optimization Problems ◽  
The Stability

Optimization problems in which the variable is not a vector but a symmetric matrix which is required to be positive semidefinite have been intensely studied in the last ten years. Part of the reason for the interest stems from the applicability of such problems to such diverse areas as designing the strongest column, checking the stability of a differential inclusion, and obtaining tight bounds for hard combinatorial optimization problems. Part also derives from great advances in our ability to solve such problems efficiently in theory and in practice (perhaps ‘or’ would be more appropriate: the most effective computational methods are not always provably efficient in theory, and vice versa). Here we describe this class of optimization problems, give a number of examples demonstrating its significance, outline its duality theory, and discuss algorithms for solving such problems.

scholarly journals On the Problem of a Linear Function Localization on Permutations

2020 ◽  
pp. 14-18
Author(s):  
G.A. Donets ◽  
V.I. Biletskyi
Keyword(s):  
Combinatorial Optimization ◽  
Linear Function ◽  
Optimization Problems ◽  
Target Function ◽  
Combinatorial Problems ◽  
Second Step ◽  
Localization Method ◽  
Combinatorial Optimization Problems ◽  
New Methods ◽  
Is Evaluation

Combinatorial optimization problems and methods of their solution have been a subject of numerous studies, since a large number of practical problems are described by combinatorial optimization models. Many studies consider approaches to and describe methods of solution for combinatorial optimization problems with linear or fractionally linear target functions on combinatorial sets such as permutations and arrangements. Studies consider solving combinatorial problems by means of well-known methods, as well as developing new methods and algorithms of searching a solution. We describe a method of solving a problem of a linear target function localization on a permutation set. The task is to find those locally admissible permutations on the permutation set, for which the linear function possesses a given value. In a general case, this problem may have no solutions at all. In the article, we propose a newly developed method that allows us to obtain a solution of such a problem (in the case that such solution exists) by the goal-oriented seeking for locally admissible permutations with a minimal enumeration that is much less than the number of all possible variants. Searching for the solution comes down to generating various permutations and evaluating them. Evaluation of each permutation includes two steps. The first step consists of function decreasing by transposing the numbers in the first n – 3 positions, and the second step is evaluation of the permutations for the remaining three numbers. Then we analyze the correlation (which is called balance) to define whether the considered permutation is the solution or not. In our article, we illustrate the localization method by solving the problem for n = 5. Keywords: localization, linear function, permutation, transposition, balance, position.

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