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134
Valeriy Dubrovin and Sergey Subbotin
Methods of Optimization and TheirApplications
in Neural Networks Learning
SUMMARY
The methods of numerical optimization are widely used in practice and examples of application of optimization methods at problem solving of artificial neural networks learning, and also at problem solving of many-dimensional non-linear approximation are considered in the book.
The book is intended for the students of speciality "Software of Automated Systems" and also can be used by the engineers, postgraduate students and students of various specialities for acquisition of practical skills of the solution of optimization problems.
Introduction contains definition of a sphere of optimization methods problems and review of areas of their application.
Chapter 1. Linear programming contains a statement of a problem of linear programming, description of it's properties, and also contains a simplex method for solution of a linear programming problem.
Chapter 2. Non-linear programming contains a statement of a common problem of non-linear programming and description of a classification of methods of non-linear programming.
Chapter 3. One-dimensional methods of non-linear optimization contains review of methods of direct searching (a division of interval on halves method, a Fibonacci method, a method of «golden section», a method of even searching), methods of searching with use of polynomial approximation (method of estimation with use of quadratic approximation and method of a series estimation with use of quadratic approximation), and also methods of searching with use of derivatives (a Newton - Rafson method, a method of mean point, a method of
135
intersecting (method of chords) and a method of searching with use of cubic approximation).
Chapter 4. Many-dimensional methods of non-linear optimization without gradients contains description of a Nelder-Meed method, a Hook-Jeeves method, and also a conjugate gradient method of Powell.
Chapter 5. Many-dimensional gradient methods of non-linear optimization describes a Cauchy’s method, a Newton’s method, algorithms of conjugate gradients of Fletcher-Reeves and Polack-Ribiere, a Partan-method, a Zautendaick’s method, a method of multiparameter searching, quasi-Newton methods, a Levenberg-Marquardt algorithm, a generalized gradient algorithm, and also an integral method of optimization.
Chapter 6. Methods of many-dimensional optimization with limitations contains description of a penal functions method and a method SUMT.
Chapter 7. Methods of optimization in approximation and neural networks learning contains review of basic concepts of the theory of artificial neural networks and formulation of problems of neural networks learning as optimization problems. In this chapter the problems of application of optimization methods for solving of approximation problem are also considered. The algorithm of cluster-regression approximation developed by the authors is proposed for construction and training of neural nretworks.
Chapter 8. Software of numerical optimization is contained a review of main means for solving of optimization problems and modelling of neural networks on the basis of Matlab software package
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Conclusions contains recommendations for application of the considered methods of optimization at the solution of practical problems.
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