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An Introduction to Continuous Optimization
Foundations and Fundamental Algorithms
- 1I Introduction
- 13Modelling and classification
- 1.13Modelling of optimization problems
- 1.211A quick glance at optimization history
- 1.313Classification of optimization models
- 1.416Conventions
- 1.518Applications and modelling examples
- 1.619On optimality conditions
- 1.720Soft and hard constraints
- 1.7.120Definitions
- 1.7.221A derivation of the exterior penalty function
- 1.823A road map through the material
- 1.928Background and didactics
- 1.1030Illustrating the theory
- 1.1131Notes and further reading
- 33II Fundamentals
- 235Analysis and algebra—A summary
- 2.135Reductio ad absurdum
- 2.236Linear algebra
- 2.339Analysis
- 343Convex analysis
- 3.143Convexity of sets
- 3.245Polyhedral theory
- 3.2.145Convex hulls
- 3.2.249Polytopes
- 3.2.350Polyhedra
- 3.2.456Fourier Elimination
- 3.2.560Farkas’ Lemma
- 3.2.663Minkowski–Weyl Theorem
- 3.365Conv ex functions
- 3.474Application: the pro jection of a vector onto a convex set
- 3.576Notes and further reading
- 79III Optimality Conditions
- 481An introduction to optimality conditions
- 4.181Local and global optimality
- 4.284Existence of optimal solutions
- 4.2.184A classic result
- 4.2.287*Non-standard results
- 4.2.389Special optimal solution sets
- 4.391Optimality conditions for unconstrained optimization
- 4.494Optimality conditions for optimization over convex sets
- 4.5102Near-optimality in convex optimization
- 4.6103Applications
- 4.6.1103Continuity of convex functions
- 4.6.2105The Separation Theorem
- 4.6.3115Euclidean pro jection
- 4.6.4116Fixed point theorems
- 4.7122Notes and further reading
- 5125Optimality conditions
- 5.1125Relations between optimality conditions and CQs at a glance
- 5.2126A note of caution
- 5.3128Geometric optimality conditions
- 5.4134The Fritz John conditions
- 5.5141The Karush–Kuhn–Tucker conditions
- 5.6145Proper treatment of equality constraints
- 5.7147Constraint qualifications
- 5.7.1147Mangasarian–Fromovitz CQ (MF CQ)
- 5.7.2148Slater CQ
- 5.7.3148Linear independence CQ (LICQ)
- 5.7.4149Affine constraints
- 5.8150Sufficiency of the KKT conditions under convexity
- 5.9152Applications and examples
- 5.10154Notes and further reading
- 6157Lagrangian duality
- 6.1157The relaxation theorem
- 6.2158Lagrangian duality
- 6.2.1158Lagrangian relaxation and the dual problem
- 6.2.2163Global optimality conditions
- 6.2.3165Strong duality for convex programs
- 6.2.4170Strong duality for linear and quadratic programs
- 6.2.5172Two illustrative examples
- 6.3174Differentiability properties of the dual function
- 6.3.1174Subdifferentiability of convex functions
- 6.3.2178Differentiability of the Lagrangian dual function
- 6.4180*Subgradient optimization methods
- 6.4.1180Convex problems
- 6.4.2186Application to the Lagrangian dual problem
- 6.4.3189The generation of ascent directions
- 6.5190*Obtaining a primal solution
- 6.5.1191Differentiability at the optimal solution
- 6.5.2192Everett’s Theorem
- 6.6194*Sensitivity analysis
- 6.6.1194Analysis for convex problems
- 6.6.2196Analysis for differentiable problems
- 6.7197Applications
- 6.7.1197Electrical networks
- 6.7.2A Lagrangian relaxation of the traveling salesman
- 201problem
- 6.8206Notes and further reading
- 209IV Linear Programming
- 7211Linear programming: An introduction
- 7.1211The manufacturing problem
- 7.2212A linear programming model
- 7.3213Graphical solution
- 7.4213Sensitivity analysis
- 7.4.1214An increase in the number of large pieces available
- 7.4.2215An increase in the number of small pieces available
- 7.4.3216A decrease in the price of the tables
- 7.5217The dual of the manufacturing problem
- 7.5.1217A competitor
- 7.5.2217A dual problem
- 7.5.3218Interpretations of the dual optimal solution
- 8219Linear programming models
- 8.1219Linear programming modelling
- 8.2224The geometry of linear programming
- 8.2.1225Standard form
- 8.2.2Basic feasible solutions and the Representation The-
- 228orem
- 8.2.3235Adjacent extreme points
- 8.3237Notes and further reading
- 9239The simplex method
- 9.1239The algorithm
- 9.1.1240A BFS is known
- 9.1.2247A BFS is not known: phase I & II
- 9.1.3251Alternative optimal solutions
- 9.2251Termination
- 9.3252Computational complexity
- 9.4253Notes and further reading
- 10255LP duality and sensitivity analysis
- 10.1255Introduction
- 10.2256The linear programming dual
- 10.2.1257Canonical form
- 10.2.2257Constructing the dual
- 10.3261Linear programming duality theory
- 10.3.1261Weak and strong duality
- 10.3.2264Complementary slackness
- 10.3.3268An alternative derivation of the dual linear program
- 10.4268The dual simplex method
- 10.5272Sensitivity analysis
- 10.5.1273Perturbations in the objective function
- 10.5.2274Perturbations in the right-hand side coefficients
- 10.5.3275Addition of a variable or a constraint
- 10.6277Column generation in linear programming
- 10.6.1277The minimum cost multi-commodity network flow problem
- 10.6.2280The column generation principle
- 10.6.3282An algorithm instance
- 10.7284Notes and further reading
- 287V Algorithms
- 11289Unconstrained optimization
- 11.1289Introduction
- 11.2291Descent directions
- 11.2.1291Introduction
- 11.2.2293Newton’s method and extensions
- 11.2.3297Least-squares problems and the Gauss–Newton al- gorithm
- 11.3299The line search problem
- 11.3.1299A characterization of the line search problem
- 11.3.2300Approximate line search strategies
- 11.4302Convergent algorithms
- 11.5305Finite termination criteria
- 11.6307A comment on non-differentiability
- 11.7309Trust region methods
- 11.8310Conjugate gradient methods
- 11.8.1310Conjugate directions
- 11.8.2311Conjugate direction methods
- 11.8.3313Generating conjugate directions
- 11.8.4313Conjugate gradient methods
- 11.8.5316Extension to non-quadratic problems
- 11.9317A quasi-Newton method: DFP
- 11.10320onvergence rates
- 11.11321mplicit functions
- 11.12322otes and further reading
- 12323Feasible-direction methods
- 12.1323Feasible-direction methods
- 12.2325The Frank–Wolfe algorithm
- 12.3328The simplicial decomposition algorithm
- 12.4331The gradient pro jection algorithm
- 12.5337Application: traffic equilibrium
- 12.5.1337Model analysis
- 12.5.2340Algorithms and a numerical example
- 12.6341Algorithms given by closed maps
- 12.6.1341Algorithmic maps
- 12.6.2344Closed maps
- 12.6.3345A convergence theorem
- 12.6.4348Additional results on composite maps
- 12.6.5352Convergence of some algorithms defined by closed descent maps
- 12.7353Active set methods
- 12.7.1354The methods of Zoutendijk, Topkis, and Veinott
- 12.7.2359A primal–dual active set method for polyhedral constraints
- 12.7.3362Rosen’s gradient pro jection algorithm
- 12.8366Reduced gradient algorithms
- 12.8.1366Introduction
- 12.8.2367The reduced gradient algorithm
- 12.8.3369Convergence analysis
- 12.9370Notes and further reading
- 13373Constrained optimization
- 13.1373Penalty methods
- 13.1.1374Exterior penalty methods
- 13.1.2378Interior penalty methods
- 13.1.3381Computational considerations
- 13.1.4382Applications and examples
- 13.2385Sequential quadratic programming
- 13.2.1385Introduction
- 13.2.2388A penalty-function based SQP algorithm
- 13.2.3393A numerical example on the MSQP algorithm
- 13.2.4394On recent developments in SQP algorithms
- 13.3394A summary and comparison
- 13.4395Notes and further reading
- 397VI Exercises
- 14399Exercises
- 399Chapter 1: Modelling and classification
- 402Chapter 3: Convexity
- 406Chapter 4: An introduction to optimality conditions
- 412Chapter 5: Optimality conditions
- 417Chapter 6: Lagrangian duality
- 422Chapter 8: Linear programming models
- 424Chapter 9: The simplex method
- 426Chapter 10: LP duality and sensitivity analysis
- 431Chapter 11: Unconstrained optimization
- 436Chapter 12: Feasible direction methods
- 439Chapter 13: Constrained optimization
- 15443Answers to the exercises
- 443Chapter 1: Modelling and classification
- 446Chapter 3: Convexity
- 449Chapter 4: An introduction to optimality conditions
- 452Chapter 5: Optimality conditions
- 455Chapter 6: Lagrangian duality
- 457Chapter 8: Linear programming models
- 459Chapter 9: The simplex method
- 462Chapter 10: LP duality and sensitivity analysis
- 464Chapter 11: Unconstrained optimization
- 466Chapter 12: Optimization over convex sets
- 467Chapter 13: Constrained optimization
- 469References
- 487Index
Information
Språk:
EngelskaISBN:
9789144115290Utgivningsår:
2005Revisionsår:
2016Artikelnummer:
32217-03Upplaga:
TredjeSidantal:
508