## An Optimization Primer: On Models, Algorithms, and DualityOptimization is the art, science and mathematics of finding the "best" member of a finite or infinite set of possible choices, based on some objective measure of the merit of each choice in the set. Three key facets of the subject are: - the construction of optimization models that capture the range of available choices within a feasible set and the measure-of-merit of any particular choice in a feasible set relative to its competitors; - the invention and implementation of efficient algorithms for solving optimization models; - a mathematical principle of duality that relates optimization models to one another in a fundamental way. Duality cuts across the entire field of optimization and is useful, in particular, for identifying optimality conditions, i.e., criteria that a given member of a feasible set must satisfy in order to be an optimal solution. This booklet provides a gentle introduction to the above topics and will be of interest to college students taking an introductory course in optimization, high school students beginning their studies in mathematics and science, the general reader looking for an overall sense of the field of optimization, and specialists in optimization interested in developing new ways of teaching the subject to their students. John Lawrence Nazareth is Professor Emeritus in the Department of Mathematics at Washington State University and Affiliate Professor in the Department of Applied Mathematics at the University of Washington. He is the author of two recent books also published by Springer-Verlag which explore the above topics in more depth, Differentiable Optimization and Equation Solving (2003) and DLP and Extensions: An Optimization Model and Decision Support System (2001). |

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### Contents

Simple Motivating Examples | 1 |

12 Watering the Garden | 2 |

13 Chopping Wood | 3 |

14 Going Fishing | 4 |

15 Summary | 5 |

A Quintessential Optimization Problem | 7 |

22 Algorithms | 8 |

23 Duality | 14 |

64 Notes | 53 |

An Algorithmic Revolution | 55 |

71 Affine Scaling | 56 |

72 Central Path | 60 |

73 InteriorPoint Algorithms | 64 |

74 Notes | 68 |

Nonlinear Programming | 69 |

81 Geometric Perspective | 70 |

24 Notes | 18 |

Duality on Bipartite Networks | 19 |

31 Matching | 21 |

32 Covering | 23 |

33 KönigEgerváry Duality | 27 |

34 Notes | 28 |

A Network Flow Overview | 29 |

42 A Network Flow Tree | 31 |

Combinatorial visŕvis Continuous | 34 |

44 Notes | 35 |

Duality in Linear Programming | 37 |

52 Farkas Duality and LP Optimality | 40 |

53 Notes | 44 |

The Golden Age of Optimization | 45 |

62 Linear Programming in Practice | 50 |

63 Network Simplex Algorithm | 52 |

82 Algebraic Perspective | 72 |

83 Information Costs | 73 |

84 Dimensions | 74 |

85 Constraints | 75 |

86 Differentiable Programming | 76 |

87 Notes | 78 |

DLP and Extensions | 79 |

92 A Rangeland Improvement Problem | 82 |

93 ResourceDecision Software | 86 |

94 Notes | 91 |

Optimization The Big Picture | 93 |

102 Notes | 97 |

References | 99 |

105 | |

About the Author | 109 |