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区间分析及其在优化中的应用.doc
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区间分析及其在优化中的应用.doc
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1
INTERVAL ANALYSIS AND ITS
APPLICATIONS TO OPTIMIZATION
IN BEHAVIOURAL ECOLOGY
by
Justin Tung
CS 490 Independent Research Report
Instructor: David Schwartz
Date: December 19, 2001
2
Table of Contents
Abstract 4
1. Introduction 5
1.1 Interval Analysis 5
1.1.1 Basics and Notation 5
1.1.2 Uncertainty and Approximating Values 5
1.1.3 Interval Arithmetic and Functions 6
1.2 Foraging Theory 7
1.2.1 Basics of Foraging Models 7
1.2.2 Simplistic Analytic Foraging Model 8
1.2.3 The Optimal Residence Time 11
2. Research Problem and Methods 12
2.1 Motivation 12
2.1.1 Problems with Fixed Point Optimization in Foraging Models 12
2.1.2 Interval Analysis as Uncertainty in Method 13
2.1.3 Research Problem 13
2.1.4 Software 14
2.2 Methodology 14
2.2.1 Fixed-Point Analysis 14
2.2.1.1 General Method 14
2.2.1.2 Algorithm: Bisection Method 15
2.2.2 Interval Analysis 16
2.2.2.1 General Method 16
2.2.2.2 Algorithm: Interval Newton’s Method 16
2.2.2.3 Variation and Constraints on Parameters 17
3. Numerical Analysis of Model 18
3.1 Fixed Point Analysis 18
3.1.1 Graphical Analysis 18
3.1.2 Optimization and Analysis 23
3.2 Interval Analysis 26
3.2.1 True Solutions and Interval Optimization 26
3.2.2 Stability Analysis 29
3
4. Conclusions and Future Exploration 30
4.1 Results of Numerical Study 30
4.1.1 Comparison of Fixed Point and Interval Roots 30
4.1.2 Applications to Foraging Model 30
4.2 Future Exploration 32
Bibliography 33
4
Abstract
Interval Analysis is a means of representing uncertainty by replacing single (fixed-point)
values with intervals. In this project, interval analysis is applied to a foraging model in
behavioural ecology. The model describes an individual foraging in a collection of
continuously renewing resource patches. This model is used to determine the optimal
residence time of the forager in a resource patch assuming the forager wants to maximize its
rate of resource intake. Before applying interval analysis, fixed-point (non-interval)
optimization will be done to serve as a basis. Certain parameters in the model will then be
replaced with intervals and interval-based optimization conducted. A comparison of the
interval and fixed-point results will be done as well as analysis of parameter intervals and
their constraints, root approximations, and applications to the model.
5
Chapter 1: Introduction
1.1 Interval Analysis
1.1.1 Basics and Notation
This paper will explain only the basics of Interval Analysis (IA) needed to
understand the topics covered and assumes some prior knowledge of IA and Matlab (see
2.1.4 regarding Matlab). For a formal mathematical introduction and in depth coverage of
concepts see Schwartz (1999) or Moore (1966) listed in the references. Interval analysis was
initially developed in the late 1960’s to bound computational error and it is a deterministic
way of representing uncertainty in values by replacing a number with a range of values
(Schwartz 17). Fixed-point analysis is simply analysis using non-interval values where there is
no uncertainty in the values. As a result, IA uncertainty concepts can be used to model
varying biological parameters in the ecological model to be explored in section 1.2 and also
to frame fixed-point results.
IA’s mathematical definitions and notations are extended from set theory and
ordered numerical sets called intervals (Schwartz 30). This paper considers closed interval
analysis with the following definitions of an interval (using Matlab upper bound, lower
bound style notation):
}),sup(),inf(),sup()inf(|{)]sup(),[inf( ������ xxxxxxxxxx
inf(x) – denotes the infinum, or lower bound of x
sup(x) – denotes the supremum or upper bound of x
1.1.2 Uncertainty and Approximating Values
There are a several useful quantities related to the concept of the interval: size,
radius, and midpoint. The size (or thickness) of an interval indicates the uncertainty in a
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