Contents. Preface CHAPTER III

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1 Optimization Edited by G.L. Nemhauser Georgia Institute of Technology A.H.G. Rinnooy Kan Erasmus University Rotterdam M.J. Todd Cornell Univerisity 1989 NORTH-HOLLAND AMSTERDAM NEW YORK OXFORD TOKYO

2 Preface v CHARTER I A View of Unconstrained Optimization J.E. Dennis Jr. and R.B. Schnabel 1 1. Preliminaries 1 2. Newton's method 8 3. Derivative approximations Globally convergent methods Non-Taylor series methods Some current research directions 58 References 66 CHAPTER II Linear Programming D. Goldfarb and M.J. Todd Introduction Geometrie interpretation The simplex method Duality and sensitivity analysis Exploiting strueture Column generation and the decomposition principle The complexity of linear programming The elipsoid method Karmarkar's projeetive scaling algorithm 141 References 165 CHAPTER III Constrained Nonlinear Programming P.E. Gill, W. Murray, M.A. Saunders and M.H. Wright Equality constraints Equality-constrained quadratic programming 176

3 XU 3. Overview of methods The quadratic penality function The / x penalty function Sequential quadratic programming methods Sequential linearly constrained methods Augmented Lagrangian methods Inequality constraints Inequality-constrained quadratic programming Penalty-function methods for inequalities Sequential quadratic programming methods Sequential linearly constrained methods Augmented Lagrangian methods Barrier-function methods 205 References 208 CHAPTER IV Network Flows R.K. Ahuja, T.L. Magnanti and J.B. Orlin Introduction Basic properties of network flows Shortest paths Maximum flows Minimum cost flows Reference notes 332 Acknowledgements 360 References 360 CHAPTER V Polyhedral Combinatorics W.R. Pulleyblank Min-max relations, NP and co-np Weighted min-max relations and polyhedra Basic theory of polyhedra and linear Systems Linear Systems and combinatorial optimization Separation and partial descriptions Polarity, blocking and antiblocking Strengthening min-max theorems I: Essential inequalities Strengthening min-max theorems II: Dual integrality Dimension Adjacency Extended formulations and projection 432 Appendix: P, NP and co-np 437 Acknowledgements 440 References

4 xiii CHAPTER VI Integer Programming G.L. Nemhauser and L.A. Wolsey Introduction Integer programming modeis Choices in model formulation Properties of integral polyhedra and computational complexity Relaxation and valid inequalities Duality Cutting plane algorithms Branch-and-bound Heuristics Notes 517 References 521 CHAPTER VII Nondifferentiable Optimization C. Lemarechal Introduction Examples of nonsmooth problems Failure of smooth methods Special methods for special problems Subgradient methods Bündle methods Directions for future developments Commented bibliography 566 Bibliography 569 CHAPTER VIII Stochastic Programming R.J.-B. Wets Introduction: The model Expectation functionals Anticipative modeis and adaptive modeis Recourse problems Optimality conditions Approximations Solution procedures Stability and incomplete Information References 623

5 XIV CHAPTER IX Global Optimization A.H.G. Rinnooy Kan and G.T. Timmer Introduction Partition and search Approximation and search Global decrease Improvement of local minima Enumeration of local minima Concluding remarks 657 References 659 CHAPTER X Multiple Criteria Decision Making: Five Basic Concepts P.L. Yu Introduction Preference structures and classes of nondominated Solutions Goal setting and compromise Solutions Value functions Nondominated Solutions and cone domination structures Linear cases and MC 2 simplex method Conclusion 697 References 697 Subject Index 701