Foreword | p. xi |
Preface | p. xiii |
The study of optimization | p. 1 |
Mathematics and practitioners | p. 1 |
Applying optimization methods | p. 3 |
Mathematical background | p. 5 |
Variables and functions | p. 5 |
Differentiation and integration of functions | p. 6 |
Vectors | p. 8 |
Definition of regions | p. 10 |
Matrices | p. 11 |
Permutations | p. 12 |
Set notation | p. 12 |
Definition of a network | p. 13 |
Probability distributions | p. 14 |
Optimization problems and methods | p. 17 |
Basic components | p. 17 |
Problem variables | p. 17 |
Objective functions | p. 18 |
Constraints | p. 19 |
Equality constraints | p. 19 |
Inequality constraints | p. 20 |
Discrete constraints | p. 21 |
Disjoint alternative regions | p. 22 |
The general optimization problem | p. 23 |
Global optima | p. 23 |
Local optima | p. 23 |
The search for the optimum | p. 24 |
Optimization methods | p. 25 |
Calculus and Lagrange multipliers | p. 29 |
The achievement of the calculus | p. 29 |
Restrictions on the use of the calculus | p. 30 |
Problems in one variable | p. 31 |
Unconstrained problems in two or more variables | p. 33 |
Equality constraints: Lagrange multipliers | p. 35 |
Sufficient conditions for local maxima and minima | p. 37 |
Proof of the Lagrange multiplier method | p. 39 |
Linear programming | p. 42 |
Linear functions | p. 42 |
The standard form | p. 43 |
Converting to the standard form | p. 43 |
The ideas of the simplex method | p. 45 |
The canonical form | p. 47 |
Improving the basis | p. 49 |
Transformation to the new canonical form | p. 51 |
Summary of the procedure for a change of basis | p. 51 |
Obtaining an initial solution | p. 54 |
The simplex tableau | p. 56 |
The revised simplex method | p. 58 |
Difficulties in special cases | p. 61 |
The fundamental theorem of linear programming | p. 61 |
The convex space of feasible solutions | p. 63 |
Optimization of non-linear functions | p. 65 |
Non-linear optimization techniques | p. 65 |
Local and global optima | p. 66 |
Finding an initial feasible solution | p. 67 |
Unconstrained problems: the failure of the calculus | p. 69 |
Steepest descent for unconstrained problems | p. 69 |
More advanced descent methods for unconstrained problems | p. 72 |
Projected gradient methods for linear constraints | p. 75 |
Created response surface techniques for general constraints | p. 80 |
Direct search procedures | p. 84 |
Separable programming | p. 88 |
Approximation programming | p. 91 |
The direction of steepest descent | p. 95 |
Construction of the projection matrix | p. 96 |
Convergence of the response surface technique | p. 97 |
Dynamic programming | p. 101 |
The aim of dynamic programming | p. 101 |
Multi-stage decision processes | p. 102 |
The dynamic programming method: forwards calculation | p. 103 |
The backwards calculation | p. 105 |
Network problems | p. 106 |
Allocation of a single resource to a number of activities | p. 108 |
Reliability problems | p. 113 |
Computational comments | p. 114 |
Allocation processes involving two types of resources or two constraints | p. 116 |
Problems with two neighbouring stages related: backward calculations | p. 118 |
A stochastic problem | p. 119 |
Branch and bound methods | p. 123 |
Solutions as tree searches | p. 123 |
Rules for the branch and bound method | p. 125 |
The three-machine scheduling problem | p. 127 |
The knapsack problem | p. 131 |
Integer programming | p. 134 |
Computational considerations | p. 138 |
Suboptimization | p. 138 |
Permutation procedures | p. 141 |
The need to suboptimize | p. 141 |
Permutation problems | p. 141 |
Locally optimal permutations | p. 143 |
Calculation of locally optimal permutations | p. 145 |
Choosing sets of exchanges | p. 149 |
Multi-permutation problems | p. 149 |
Heuristic techniques | p. 152 |
The need for skilful guessing | p. 152 |
The nature of allocation problems | p. 153 |
Problem specification and mathematical treatment | p. 161 |
Flexibility in problem formulation | p. 161 |
The choice of the problem variables | p. 162 |
Transferring factors between the objective and constraint functions | p. 162 |
Introducing auxiliary variables | p. 164 |
Transformations | p. 166 |
Duality in linear problems | p. 168 |
Interpretation of the dual problem | p. 172 |
Proof of the duality theorem | p. 174 |
Answers to exercises | p. 178 |
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