Neural Networks in Optimization

Front Cover
Springer Science & Business Media, Oct 31, 2000 - Business & Economics - 367 pages
People are facing more and more NP-complete or NP-hard problems of a combinatorial nature and of a continuous nature in economic, military and management practice. There are two ways in which one can enhance the efficiency of searching for the solutions of these problems. The first is to improve the speed and memory capacity of hardware. We all have witnessed the computer industry's amazing achievements with hardware and software developments over the last twenty years. On one hand many computers, bought only a few years ago, are being sent to elementary schools for children to learn the ABC's of computing. On the other hand, with economic, scientific and military developments, it seems that the increase of intricacy and the size of newly arising problems have no end. We all realize then that the second way, to design good algorithms, will definitely compensate for the hardware limitations in the case of complicated problems. It is the collective and parallel computation property of artificial neural net works that has activated the enthusiasm of researchers in the field of computer science and applied mathematics. It is hard to say that artificial neural networks are solvers of the above-mentioned dilemma, but at least they throw some new light on the difficulties we face. We not only anticipate that there will be neural computers with intelligence but we also believe that the research results of artificial neural networks might lead to new algorithms on von Neumann's computers.
 

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Contents

PRELIMINARIES
3
12 Pidgin Algol Language
5
13 Vector and Matrices
6
14 Convex Set and Convex Function
12
15 Digraph and Network
14
16 Algorithm Complexity and Problem Complexity
16
17 Concepts of Ordinary Differential Equations
24
18 Markov Chain
28
63 Multilayer Perceptrons and Extensions
107
64 BackPropagation
117
65 Optimization Layer by Layer
122
66 Local Solution Effect
130
FEEDBACK NEURAL NETWORKS
137
71 Convergece Analysis for discrete Feedback Networks
140
72 Discrete Hopfield Net as Contentaddressable Memory
153
73 Continuous Feedback Networks
163

INTRODUCTION TO MATHEMATICAL PROGRAMMING
31
22 Classical Algorithms for LP
38
23 Basics of Nonlinear Programming
40
24 Convex Programming
45
25 Quadratic Programming and SQPM
46
26 Duality in Nonlinear Programming
48
UNCONSTRAINED NONLINEAR PROGRAMMING
53
31 Newton Method
54
32 Gradient Method
55
33 QuasiNewton Method
59
34 Conjugate Gradient Method
60
35 Trust Region Method for Unconstrained Problems
62
CONSTRAINED NONLINEAR PROGRAMMING
65
41 Exterior Penalty Method
66
42 Interior Penalty Method
67
43 Exact Penalty Method
69
44 Lagrangian Multiplier Method
74
45 Projected Lagrangian Methods
77
46 Trust Region Method for Constrained Problem
79
BASIC ARTIFICIAL NEURAL NETWORK MODELS
81
INTRODUCTION TO ARTIFICIAL NEURAL NETWORK
83
51 What Is an Artificial Neuron?
84
52 Feedforward and Feedback StructuresChapter
90
FEEDFORWARD NEURAL NETWORKS
95
62 Simple Perceptron
101
SELFORGANIZED NEURAL NETWORKS
177
82 Competitive Learning Network Kohonen Network
179
83 Convergence Analysis
185
NEURAL ALGORITHMS FOR OPTIMIZATION
197
NN MODELS FOR COMBINATORIAL PROBLEMS
199
92 Complexity Analysis
205
93 Solving TSP by Neural Networks
207
94 NN Models for Four Color Map Problem
226
95 NN Models for Vertex Cover Problem
234
96 Discussion
236
NN FOR QUADRATIC PROGRAMMING PROBLEMS
243
101 Simple Limiter Neural Nets for QP
245
102 Saturation Limiter Neural Nets for QP
249
103 Sigmoid Limiter Neural Nets for QP
255
104 Integrator Neural Nets for QP
261
NN MODELS FOR GENERAL NONLINEAR PROGRAMMING
273
NN MODELS FOR LINEAR PROGRAMMING
289
122 Hard Limiter Neural Nets for LP
301
123 Sigmoid Limiter Neural Nets for LP
307
124 Integrator Neural Network for LP
315
A REVIEW ON NN FOR CONTINUOUS OPTIMIZATION
319
132 Some New Network Models Motivatedd by the Framework
329
References
335
Index
363
Copyright

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Page 339 - An Accelerated Learning Algorithm for Multilayer Perceptrons: Optimization Layer by Layer,
Page 349 - Y. MEYER, Wavelets and operators, Proceedings of the Special year in modern Analysis, Urbana 1986/87, published by Cambridge University Press, 1989. See also Y. MEYER, Ondelettes et Operatturs, Hermann, Paris, 1990.