Introduction to the Theory of Optimization in Euclidean Space

335 Pages·2019·20.887 MB·English

by  Samia Challal

Introduction to the
Theory of
Optimization in
Euclidean Space
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Samia Challal
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Introduction to the
Theory of
Optimization in
Euclidean Space
Samia Challal
Glendon College-York University
Toronto, Canada
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To my parents

Contents
Preface ix
Acknowledgments xi
Symbol Description xiii
Author xv
1 Introduction 1
1.1 Formulation of Some Optimization Problems . . . . . . . . . 1
1.2 Particular Subsets of Rn . . . . . . . . . . . . . . . . . . . . 8
1.3 Functions of Several Variables . . . . . . . . . . . . . . . . . 20
2 Unconstrained Optimization 49
2.1 Necessary Condition . . . . . . . . . . . . . . . . . . . . . . . 49
2.2 Classification of Local Extreme Points . . . . . . . . . . . . . 71
2.3 Convexity/Concavity and Global Extreme Points . . . . . . 93
2.3.1 Convex/Concave Several Variable Functions . . . . . 93
2.3.2 Characterization of Convex/Concave C1 Functions . . 95
2.3.3 Characterization of Convex/Concave C2 Functions . . 98
2.3.4 Characterization of a Global Extreme Point . . . . . . 102
2.4 Extreme Value Theorem . . . . . . . . . . . . . . . . . . . . 117
3 Constrained Optimization-Equality Constraints 135
3.1 Tangent Plane . . . . . . . . . . . . . . . . . . . . . . . . . . 137
3.2 Necessary Condition for Local Extreme
Points-Equality Constraints . . . . . . . . . . . . . . . . . . . 151
3.3 Classification of Local Extreme Points-Equality
Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
3.4 Global Extreme Points-Equality Constraints . . . . . . . . . 187
4 Constrained Optimization-Inequality Constraints 203
4.1 Cone of Feasible Directions . . . . . . . . . . . . . . . . . . . 204
4.2 Necessary Condition for Local Extreme Points/
Inequality Constraints . . . . . . . . . . . . . . . . . . . . . . 220
4.3 Classification of Local Extreme Points-Inequality
Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
vii
viii Contents
4.4 Global Extreme Points-Inequality Constraints . . . . . . . . 271
4.5 Dependence on Parameters . . . . . . . . . . . . . . . . . . . 292
Bibliography 315
Index 317
Preface
The book is intended to provide students with a useful background in opti-
mization in Euclidean space. Its primary goal is to demystify the theoretical
aspect of the subject.
Inpresentingthematerial,wereferfirsttotheintuitiveideainonedimension,
then make the jump to n dimension as naturally as possible. This approach
allows the reader to focus on understanding the idea, skip the proofs for later
and learn to apply the theorems through examples and solving problems. A
detailed solution follows each problem constituting an image and a deepening
of the theory. These solved problems provide a repetition of the basic princi-
ples, an update on some difficult concepts and a further development of some
ideas.
Studentsaretakenprogressivelythroughthedevelopmentoftheproofswhere
they have the occasion to practice tools of differentiation (Chain rule, Taylor
formula) for functions of several variables in abstract situation. They learn to
applyimportantresultsestablishedinadvancedAlgebraandAnalysiscourses,
like,Farkas-MinkowskiLemma,theimplicitfunctiontheoremandtheextreme
value theorem.
The book starts, in Chapter 1, with a short introduction to mathematical
modeling leading to formulation of optimization problems. Each formulation
involves a function and a set of points. Thus, basic properties of open, closed,
convex subsets of Rn are discussed. Then, usual topics of differential calculus
for functions of several variables are reminded.
Inthefollowingchapters,thestudyisdevotedtotheoptimisationofafunction
ofseveralvariablesf overasubsetS ofRn.Dependingontheparticularityof
thisset,threesituationsareidentified.InChapter2,thesetS hasanonempty
interior;inChapter3,S isdescribedbyanequationg(x)=0andinChapter 4
ix