Introduction to Optimization

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Springer Science & Business Media, Nov 3, 2003 - Business & Economics - 245 pages
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Mathematicsisplayinganevermoreimportantroleinthephysicaland biologicalsciences,provokingablurringofboundariesbetweenscientific disciplinesandaresurgenceofinterestinthemodernaswellastheclassical techniquesofappliedmathematics. Thisrenewalofinterest,bothin- searchandteaching,hasledtotheestablishmentoftheseriesTextsin AppliedMathematics(TAM). Thedevelopmentofnewcoursesisanaturalconsequenceofahighlevel ofexcitementontheresearchfrontierasnewertechniques,suchasnume- calandsymboliccomputersystems,dynamicalsystems,andchaos,mix withandreinforcethetraditionalmethodsofappliedmathematics. Thus, thepurposeofthistextbookseriesistomeetthecurrentandfutureneeds oftheseadvancesandtoencouragetheteachingofnewcourses. TAMwillpublishtextbookssuitableforuseinadvancedundergraduate andbeginninggraduatecourses,andwillcomplementtheAppliedMat- maticalSciences(AMS)series,whichwillfocusonadvancedtextbooksand research-levelmonographs. Pasadena,California J. E. Marsden Providence,RhodeIsland L. Sirovich CollegePark,Maryland S. S. Antman Preface This book should serve as an undergraduate text to introduce students of s- ence and engineering to the fascinating ?eld of optimization. Several features have been united: conciseness and completeness, brevity and clarity, emphasis on the justi?cation of ideas and techniques and also on applications, etc. One of the novelties of the text is that it ties together ?elds that are often treated as separate. Indeed, it is hard to ?nd a single textbook where mathematical p- gramming, variational problems, and optimal control problems are explained and integrated as a unity. Thus, our readers may gain an overall view of all aspects of optimization. It is also true that each of the chapters is but a timid introduction to such broad subjects as linear programming, nonlinear programming, numerical op- mizationalgorithms,variationalproblems,dynamicprogramming,andoptimal control. As a primer in optimization, our aim with this text is no more than to provide a succinct introduction to those worlds, presented in a single resource reference. Thistextcannotanddoesnotpretendtosubstituteintheleastother vii viii Preface moreprofoundtextbooksonthosesub?eldsofoptimization. Readerswithsome experience in optimization seeking a more specialized source in some of those parts will have to look for other references. Real-world applications are also far from this introduction to the subject.
  

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Contents

Introduction
1
2 THE MATHEMATICAL SETTING
6
3 THE VARIETY OF OPTIMIZATION PROBLEMS
14
4 EXERCISES
15
Linear Programming
23
2 THE SIMPLEX METHOD
30
3 DUALITY
43
4 SOME PRACTICAL ISSUES
49
6 FINAL REMARKS
131
7 EXERCISES
132
Variational Problems and Dynamic Programming
137
EXAMPLES
140
JUSTIFICATION
153
4 NATURAL BOUNDARY CONDITIONS
157
5 VARIATIONAL PROBLEMS UNDER INTEGRAL AND POINTWISE RESTRICTIONS
160
6 SUMMARY OF RESTRICTIONS FOR VARIATIONAL PROBLEMS
168

5 INTEGER PROGRAMMING
59
6 EXERCISES
63
Nonlinear Programming
67
2 LAGRANGE MULTIPLIERS
69
3 KARUSH KUHNTUCKER OPTIMALITY CONDITIONS
79
4 CONVEXITY
86
5 SUFFICIENCY OF THE KKT CONDITIONS
95
6 DUALITY AND CONVEXITY
102
7 EXERCISES
107
Approximation Techniques
111
2 LINE SEARCH METHODS
113
3 GRADIENT METHODS
116
4 CONJUGATE GRADIENT METHODS
120
5 APPROXIMATION UNDER CONSTRAINTS
124
7 VARIATIONAL PROBLEMS OF DIFFERENT ORDER
173
BELLMANS EQUATION
178
9 SOME BASIC IDEAS ON THE NUMERICAL APPROXIMATION
185
10 EXERCISES
190
Optimal Control
195
2 MULTIPLIERS AND THE HAMILTONIAN
197
3 PONTRYAGIN S PRINCIPLE
204
4 ANOTHER FORMAT
224
5 SOME COMMENTS ON THE NUMERICAL APPROXIMATION
226
6 EXERCISES
232
References
237
Index
241
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