4y Springer NONLINEAR INTEGER PROGRAMMING

Transcription

1 NONLINEAR INTEGER PROGRAMMING DUAN LI Department of Systems Engineering and Engineering Management The Chinese University of Hong Kong Shatin, N. T. Hong Kong XIAOLING SUN Department of Mathematics Shanghai University Baoshan, Shanghai P. R. China 4y Springer

2 Contents Dedication List of Figures ListofTables Preface Acknowledgments v xiii xvi xix xxii 1. INTRODUCTION Classification of Nonlinear Integer Programming Formulations Examples of Applications Resource allocation in production planning Portfolio selection Redundancy optimization in reliability networks Chemical engineering Difficulties and Challenges Organization of the Book Notes OPTIMALITY, RELAXATION AND GENERAL SOLUTION PROCEDURES Optimality Condition via Bounds Partial Enumeration Outline of the general branch-and-bound method The back-track scheme Continuous Relaxation and Lagrangian Relaxation Continuous relaxation Lagrangian relaxation 25

3 viii NONLINEAR INTEGER PROGRAMMING Continuous bound versus Lagrangian bound Proximity between Continuous Solution and Integer Solution Linear integer program Linearly constrained separable convex integer program Unconstrained convex integer program Penalty Function Approach Optimality Conditions for Unconstrained Binary Quadratic Problems General case Convex case Notes 44 LAGRANGIAN DUALITY THEORY Lagrangian Relaxation and Dual Formulation Dual Search Methods Subgradient method Outer Lagrangian linearization method Bündle method Perturbation Function Optimal Generating Multiplier and Optimal Primal-Dual Pair Solution Properties of the Dual Problem Lagrangian Decomposition via Copying Constraints General Lagrangian decomposition schemes quadratic case Notes 95 SURROGATE DUALITY THEORY Conventional Surrogate Dual Method Surrogate dual and its properties Surrogate dual search Nonlinear Surrogate Dual Method Notes 112 NONLINEAR LAGRANGIAN AND STRONG DUALITY Convexification and Nonlinear Support: p-th power Nonlinear Lagrangian Formulation Nonlinear Lagrangian Theory Using Equivalent Reformulation Nonlinear Lagrangian Theory Using Logarithmic-Exponential Dual Formulation 127

4 Contents ix 5.4 Generalized Nonlinear Lagrangian Theory for Singly-Constrained Nonlinear Integer Programming Problems Notes 148 NONLINEAR KNAPSACK PROBLEMS Continuous-Relaxation-Based Branch-and-Bound Methods Multiplier search method Pegging method Linearization Method linearization Algorithms for 0-1 linear knapsack problem Convergent Lagrangian and Domain Cut Algorithm Derivation of the algorithm Domain cut The main algorithm Multi-dimensional nonlinear knapsack problems Concave Nonlinear Knapsack Problems Linear approximation Domain cut and linear approximation method Reliability Optimization in Series-Parallel Reliability Networks Maximal decreasing property Implementation and Computational Results Test problems Heuristics for feasible Solutions Numerical results of Algorithm 6.2 for singly constrained cases Numerical results of Algorithm 6.2 for multiply constrained cases Numerical results of Algorithm Comparison results Notes 206 SEPARABLE INTEGER PROGRAMMING Dynamic Programming Method Backward dynamic programming Forward dynamic programming Singly constrained case Hybrid Method 217

5 X NONLINEAR INTEGER PROGRAMMING Dynamic programming procedure Incorporation of elimination procedure Relaxation of (RSP k ) Convergent Lagrangian and Objective Level Cut Method Motivation Algorithm description Implementation of dynamic programming Computational experiment Notes NONLINEAR INTEGER PROGRAMMING WITH A QUADRATIC OBJECTIVE FUNCTION Quadratic Contour Cut Ellipse of quadratic contour Contour cuts of quadratic function Convergent Lagrangian and Objective Contour Cut Method Extension to Problems with Multiple Constraints Extension to Problems with Indefinite q Computational Results Test problems Computational results Comparison with other methods Note NONSEPARABLE INTEGER PROGRAMMING Branch-and-Bound Method based on Continuous Relaxation Branching variables Branching nodes Lagrangian Decomposition Method Monotone Integer Programming Discrete polyblock method for (MIP) Convexity and monotonicity Equivalent transformation using convexification Polyblock and convexification method for (MIP) Computational results Notes 290

6 Contents XI 10. UNCONSTRAINED POLYNOMIAL 0-1 OPTIMIZATION RoofDuality Basic concepts Relation to other linearization formulations Quadratic case LocalSearch Basic Algorithm Continuous Relaxation and its Convexification Unconstrained Quadratic 0-1 Optimization A polynomially solvable case Equivalence to maximum-cut problem Variable fixation Notes CONSTRAINED POLYNOMIAL 0-1 PROGRAMMING Reduction to Unconstrained Problem Linearization Methods Branch-and-Bound Method Upper bounds and penalties Branch-and-bound method Cutting Plane Methods Generalized covering relaxation Lower bounding linear function Linearization of polynomial inequality Quadratic 0-1 Knapsack Problems Lagrangian dual of {QKP) Heuristics for finding feasible Solutions Branch-and-bound method Alternative upper bounds Notes TWO LEVEL METHODS FOR CONSTRAINED POLYNOMIAL 0-1 PROGRAMMING Revised Taha's Method Definitions and notations Fathoming, consistency and augmentation Solution algorithm 357

7 xii NONLINEAR INTEGER PROGRAMMING 12.2 Two-Level Method for p-norm Surrogate Constraint Formulation Convergent Lagrangian Method Using Objective Level Cut Computational Results Notes MIXED-INTEGER NONLINEAR PROGRAMMING Introduction Branch-and-Bound Method Generalized Benders Decomposition Outer Approximation Method Nonconvex Mixed-Integer Programming Convex relaxation Convexification method Notes GLOBAL DESCENT METHODS Local Search and Global Descent Local minima and local search Identification of global minimum from among local minima A Class of Discrete Global Descent Functions Condition(Dl) Condition (D2) Condition (D3) The Discrete Global Descent Method Computational Results Notes 416 References 419 Index 433