Generalized Convexity and Optimization

Theory and Applications

By: Alberto Cambini, Laura Martein

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Paperback | 15 October 2008

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In the latter part of the twentieth century, the topic of generalizations of convexfunctions has attracted a sizable number of researchers, both in ma- ematics and in professional disciplines such as economics/management and engineering. In 1994 during the 15th International Symposium on Mathem- ical Programming in Ann Arbor, Michigan, I called together some colleagues to start an a?liation of researchers working in generalized convexity. The international Working Group of Generalized Convexity (WGGC) was born. Its website at www.genconv.org has been maintained by Riccardo Cambini, University of Pisa. Riccardo s father, Alberto Cambini, and Alberto s long-term colleague Laura Martein in the Faculty of Economics, University of Pisa, are the - authors of this volume. My own contact with generalized convexity in Italy datesbacktomy?rstvisittotheirdepartmentin1980, atatimewhenthe?rst international conference on generalized convexity was in preparation. Thirty years later it is now referred to as GC1, an NATO Summer School in V- couver, Canada. Currently WGGC is preparing GC9 which is to take place in Kaohsiung, Taiwan. As founding chair and also current chair of WGGC, I am delighted to see the continued interest in generalized convexity of functions, augmented by the topic of generalized monotonicity of maps. Eight international conferences have taken place in this research area, in North America (2), Europe (5) and Asia (1). We thought it was now time to return to Asia since our membership has shifted towards Asia. AsanappliedmathematicianIhavetaughtmostlyinmanagementschools."

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"The optimization of generalized convex functions plays an important role in many applications, especially in economics. The book gives an introduction to the theory and the application of such functions and their use in optimization. ... The book is self-contained and includes many exercises with solutions. I highly recommend this book to researchers and graduate students interested in optimization and their applications." (Walter Alt, Zentralblatt MATH, Vol. 1157, 2009)

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You Can Find This Book In

Non-Fiction Mathematics Optimisation Linear Programming Business & Management Operational Research Economics Microeconomics Applied Mathematics

Convex Functions	p. 1
Introduction	p. 1
Convex Sets	p. 1
Topological Properties of Convex Sets	p. 3
Relative Interior of Convex Sets	p. 4
Extreme Points and Extreme Directions	p. 4
Supporting Hyperplanes and Separation Theorems	p. 6
Convex Cones and Polarity	p. 7
Convex Functions	p. 10
Algebraic Structure of the Convex Functions	p. 13
Composite Function	p. 13
Differentiable and Twice Differentiable Convex Functions	p. 14
Convexity and Homogeneity	p. 16
Minima of Convex Functions	p. 17
Exercises	p. 18
References	p. 21
Non-Differentiable Generalized Convex Functions	p. 23
Introduction	p. 23
Quasiconvexity and Strict Quasiconvexity	p. 23
Semistrict Quasiconvexity	p. 30
Generalized Convexity of Some Homogeneous Functions	p. 34
The Cobb-Douglas Function	p. 34
The Constant Elasticity of Substitution (C.E.S.) Function	p. 35
The Leontief Production Function	p. 35
A Generalized Cobb-Douglas Function	p. 35
Generalized Quasiconvex Functions in One Variable	p. 36
Exercises	p. 38
References	p. 40
Differentiable Generalized Convex Functions	p. 41
Introduction	p. 41
Differentiable Quasiconvex and Pseudoconvex Functions	p. 41
Differentiable Quasiconvex Functions	p. 41
Pseudoconvex Functions	p. 43
Relationships	p. 48
Quasilinearity and Pseudolinearity	p. 50
Quasilinearity and Semistrict Quasilinearity	p. 50
Pseudolinearity	p. 53
Twice Differentiable Generalized Convex Functions	p. 57
Quasiconvex Functions	p. 57
Pseudoconvex Functions	p. 60
Characterizations in Terms of the Bordered Hessian	p. 61
Generalized Convexity at a Point	p. 63
Exercises	p. 68
References	p. 71
Optimality and Generalized Convexity	p. 73
Introduction	p. 73
Necessary Optimality Conditions Via Separation Theorems	p. 73
Generalized Convexity and Constraint Qualifications	p. 78
Sufficiency of the Karush-Kuhn-Tucker Conditions	p. 81
Local-Global Property	p. 82
Maxima and Generalized Convexity	p. 84
Minima, Maxima and Pseudolinearity	p. 86
Economic Applications	p. 87
The Utility Maximization Problem	p. 89
The Expenditure Minimization Problem	p. 91
The Profit Maximization Problem and the Cost Minimization Problem	p. 91
Invex Functions	p. 93
Exercises	p. 95
References	p. 97
Generalized Convexity and Generalized Monotonicity	p. 99
Introduction	p. 99
Concepts of Generalized Monotonicity	p. 99
Differentiable Generalized Monotone Maps	p. 103
Generalized Monotonicity of Maps of One Variable	p. 103
Generalized Monotonicity of Affine Maps	p. 105
Relationships Between Generalized Monotonicity and Generalized Convexity	p. 108
The Generalized Charnes-Cooper Transformation	p. 110
References	p. 112
Generalized Convexity of Quadratic Functions	p. 115
Introduction	p. 115
Preliminary Results	p. 115
Some Properties of a Quadratic form Associated with a Symmetric Matrix Having One Simple Negative Eigenvalue	p. 116
Quadratic Functions	p. 119
Quadratic Functions of Non-negative Variables	p. 126
Pseudoconvexity on a Closed Set	p. 128
Pseudoconvexity on the Non-negative Orthant	p. 130
Generalized Convexity of a Quadratic form on R[superscript 2 subscript +]	p. 131
A Special Case	p. 132
Exercises	p. 134
References	p. 135
Generalized Convexity of Some Classes of Fractional Functions	p. 137
Introduction	p. 137
The Ratio of a Quadratic and an Affine Function	p. 137
The Sum of a Linear and a Linear Fractional Function	p. 141
Pseudoconvexity and the Charnes-Cooper Variable Transformation	p. 147
Sum of Two Linear Fractional Functions	p. 149
Exercises	p. 154
References	p. 157
Sequential Methods for Generalized Convex Fractional Programs	p. 159
Introduction	p. 159
The Linear Fractional Problem	p. 160
Isbell-Marlow's Algorithm	p. 162
Charnes-Cooper's Algorithm	p. 164
Martos' Algorithm	p. 166
Cambini-Martein's Algorithm	p. 168
The Case of an Unbounded Feasible Region	p. 172
A Generalized Linear Fractional Problem	p. 175
Sequential Methods	p. 176
The Sum of Two Linear Fractional Functions	p. 182
Generalized Linear Multiplicative Programs	p. 183
The Sum of a Linear Function and the Product of Two Affine Functions	p. 183
The Product Between an Affine Function and the Power of an Affine Function	p. 185
The Optimal Level Solutions Method	p. 189
References	p. 193
Solutions	p. 195
References	p. 213
Mathematical Review	p. 229
Sets	p. 229
The Euclidean Space R[superscript n]	p. 230
Topological Concepts	p. 234
Functions	p. 235
Concave and Generalized Concave Functions	p. 241
Index	p. 247
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Generalized Convexity and Optimization

Theory and Applications

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Paperback

Industry Reviews

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