Neural Networks in Optimization pp 3-29 | Cite as

# Preliminaries

## Abstract

In this book letters *x*,*y*, ... are scalars (real numbers) and the letters in bold italics, *x*, *y*, ... , stand for column vectors in the *n*-dimensional *Euclidian space* ℝ^{ n } unless explicitly stated otherwise. The *i*-th *component* or *element* of a vector *x* is denoted by *x* _{ i }. The superscripts usually represent different vectors, for example, *x* ^{ k },*k* ∈ *K* where *K* is a set of indices. In general the columns of an *m x n* matrix *A* (or *A* _{ m×n }) are denoted by *a* ^{1},*a* ^{2}
⋯,*a* ^{ n }, while the rows by *A* _{1}, *A* _{2}, ..., *A* _{ m } or {* A*}

_{i},

*i*= 1, ⋯ ,

*m*. The entry in row i and column

*j*of a matrix

*is denoted by*

**A***a*

_{ ij }.

*g*(

*x*)is a column vector-valued function with scalar-valued functions

*g*

_{ 1 }(

*x*),

*g*

_{ 2 }(

*x*), ... ,

*g*

_{ m }(

*x*) as its components. In the following chapters, we sometimes write

*f*

^{ k },

*g*

^{ k }as abbreviations for

*f*(

*x*

^{ k }),

*g*(

*x*

^{ k }).

## Keywords

Equilibrium Point Convex Function Lyapunov Function Travel Salesman Problem Hamilton Path## Preview

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