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Bonnans F.J., Gilbert C.J., Lemarechal C. — Numerical Optimization
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Название: Numerical OptimizationАвторы: Bonnans F.J., Gilbert C.J., Lemarechal C.Аннотация: Numerical Optimization has numerous applications in engineering sciences, operations research, economics, finance, etc. Starting with illustrations of this ubiquitous character, this book is essentially devoted to numerical algorithms for optimization, which are exposed in a tutorial way. It covers fundamental algorithms as well as more specialized and advanced topics for unconstrained and constrained problems. The theoretical bases of the subject, such as optimality conditions, Lagrange multipliers or duality, although recalled, are assumed known. Most of the algorithms described in the book are explained in a detailed manner, allowing straightforward implementation. This level of detail is intended to familiarize the reader with some of the crucial questions of numerical optimization: how algorithms operate, why they converge, difficulties that may be encountered and their possible remedies. Theoretical aspects of the approaches chosen are also addressed with care, often using minimal assumptions.
Язык: Рубрика: Математика /Оптимизация и управление /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 2003Количество страниц: 418Добавлена в каталог: 22.04.2005Операции: Положить на полку |
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157 157 154 161 163 13 163 222 Adjoint equation 18 Adjoint state 18 Admissibility of the unit stepsize 48 63 253 264 278 Admissible see "Set point" Aggregation 116 119 122 Algorithm see also "Method" Algorithm bundle, dual 126 Algorithm bundle, general 118 Algorithm bundle, penalized 121 Algorithm level 120 Algorithm, Bunch & Kaufman 185 Algorithm, conjugate gradient 84 86 Algorithm, conjugate gradient, preconditioned 76 Algorithm, conjugate gradient, truncated 246 249 Algorithm, cutting-planes 112 Algorithm, descent 102 Algorithm, ellipsoid 111 Algorithm, Karmarkar 389 Algorithm, largest step 367 Algorithm, largest step, with safeguard 367 Algorithm, Newton see "Newton's algorithm" Algorithm, piecewise-linear search (PLS) 278 Algorithm, predictor-corrector 331 Algorithm, quasi-Newton see "Quasi-Newton or secant (algorithm)" Algorithm, simplex 302 Algorithm, subgradient 108 Algorithm, Uzawa 186 Analytic center 325 369 Armijo 47 80 84 118 241 250 251 Automatic differentiation see "Computational differentiation" Auxiliary problem principle 148 Basis matrix 302 BFGS 55 58 75 85 86 267 Bisection 40 Black Box 10 108 Bracket 39 Bunch & Kaufman 185 Bunch & Parlett 53 Bundle 117 123 Bundle, compression of 128 Canonical form of a linear complementarity problem 315 Central path 309 310 Central path, perturbed 346 Choleski 53 Code nlcv2 131 Coercive 100 148 Combinatorial optimization 3 10 142 Complementarity condition 141 160 Complementarity problem, linear 309 Complementarity, strict 161 315 complexity 383 Component of the Newton step, longitudinal 177 Component of the Newton step, transversal 177 Computational differentiation 18 20 89 Cone 162 Cone, critical 162 conjugate see "Algorithm direction" Constraint 3 Constraint equality 3 Constraint inequality 3 Constraint qualification 139 142 Constraint qualification, (A-CQ) 161 Constraint qualification, (LI-CQ) 161 Constraint qualification, (MF-CQ), of Mangasarian-Fromovitz 161 Constraint qualification, (S-CQ), of Slater 161 Constraint, active 4 152 154 Constraint, strongly active 161 Constraint, weakly active 161 Control see also "Adjoint" Control problem 4 7 8 16 178 control variable 7 16 Convergence in p steps 164 Convergence of bundle 133 134 Convergence of cutting-planes 113 Convergence of subgradients 109 Convergence, global 14 26 45 52 53 74 243 252 Convergence, linear 14 163 Convergence, local 166 182 183 195 207 210 Convergence, quadratic 14 164 201 Convergence, speed of 14 33 51 86 88 Convergence, superlinear 14 163 213 Convex see "Set function problem" Correction, Powell 270 274 Correction, second-order 256 Critical see "Cone direction point" Cubic fit 40 Curvature condition 267 Davidon, Fletcher & Powell 55 Decomposition of 178 Decomposition of 180 Decomposition of 179 201 Decomposition price 137 144 Decomposition price, algorithm 143 Decomposition resource 137 Decomposition, Benders 147 Decomposition, Dantzig — Wolfe 145 Dennis & More 60 63 82 Dilation 110 Direction 12 85 Direction of 107 Direction of descent 29 75 97 108 235 264 Direction of steepest descent 30 103 105 Direction, affine 318 Direction, centralization 318 Direction, conjugate 69 74 Direction, critical 162 Direction, quasi-negative curvature 246 Directional derivative 95 158 Disaggregation 145 Divergent series 110 Duality 294 Duality, gap 98 139 Duality, weak 140 146 Elliptic see "function" Equivalent sequences 163 Everett 139 Existence of solution 159 Existence of solution, dual 138 Extrapolation 40 47 feasible see "Set point" Finite difference 101 Finsler 230 Fletcher (initialization of) 39 48 Fletcher — Reeves 73 Fromovitz see "Constraint qualification" Function penalty 217 Function, affine 4 26 83 Function, convex 25 26 57 Function, convex-concave 297 Function, dual 98 138 Function, elliptic 27 48 53 74 Function, exact 218 Function, inf-compact 25 Function, lower semi-continuous 25 Function, merit 13 29 37 79 217 Function, strongly 27 61 63 65 67 Gauss — Newton 83 86 Gauss — Seidel 29 Generalized Cauchy — Schwarz inequality 225 Globalization of an algorithm 217 Globalization of an algorithm by line-search 52 235 Globalization of an algorithm by trust regions 77 Gradient 3 Gradient, reduced 187 Group 313 Growth condition 27 33 46 hessian 3 Hessian of the Lagrangian 181 Identification see "Parameter identification" Inertia of a matrix 200 Inf-compact see "Function" Instability of cutting-planes 115 Instability of steepest-descent 104 Interpolation 40 242 251 Invariant 313 Karush, Kuhn, and Tucker (KKT) see "Optimality conditions" Lagrange multiplier see "Multiplier" Lagrangian 11 78 98 160 220 294 Lagrangian relaxation 10 98 Lagrangian, augmented 100 142 222 231 272 Length of a linear problem 384 Levenberg — Marquardt 83 Line-search see also "Search" 12 72 77 78 109 Line-search, Armijo 241 251 Line-search, backtracking 242 250 251 270 Line-search, nonmonotone 263 Line-search, piecewise (PLS) 278 Line-search, watchdog 263 Line-search, Wolfe see also "Wolfe conditions" 268 Linear complementarity problem 307 Linear complementarity problem, canonical form 315 Linear complementarity problem, monotone 309 Linear complementarity problem, standard form 314 Linearization error 123 Local minimum see "Solution" Mangasarian see "Constraint qualification" Maratos effect 254 271 master program 137 143 146 Matrix, basis 176 Matrix, right inverse 176 201 method see also "Algorithm multiplier" Method, local 170 Method, primal-dual 171 Minimax (finite) 99 Minimax (infinite) 99 Minimum see "Solution" Model 12 52 77 85 117 Model, cutting-planes 111 123 Model, cutting-planes, aggregated 128 Model, enriching 118 Modified field 322 323 352 Monotone 309 Multifunction 12 Multifunction, closed 13 106 Multifunction, locally bounded 13 Multifunction, outer semi-continuous 13 Multiplier 98 142 145 Multiplier first-order 189 Multiplier method 142 Multiplier, Lagrange 11 89 160 298 Multiplier, least-squares 183 189 201 Multiplier, second-order 189 Neighborhood 331 Neighborhood, large 311 340 Neighborhood, small 311 332 Newton's algorithm 39 79 Newton's algorithm for equality constrained problems, primal version 183 Newton's algorithm for equality constrained problems, primal-dual version 175 Newton's algorithm for equality constrained problems, reduced Hessian 193 Newton's algorithm for equality constrained problems, simplified Newton 194 Newton's algorithm for inequality constrained problems see "Sequential quadratic" Newton's algorithm programming for nonlinear equations 51 165 Newton's algorithm programming for unconstrained optimization 167 Nominal decrease 112 126 Norm 263 Norm associated with a scalar product 164 Norm, dual 225 232 Objective function 3 13 16 Optimal control see "Control" Optimal partition 302 315 Optimality conditions necessary, 1st order (NC1) 26 Optimality conditions necessary, 2nd order (NC2) 26 162 Optimality conditions necessary, Karush, Kuhn, and Tucker (KKT) 160 Optimality conditions necessary, reduced 190 Optimality conditions sufficient, 2nd order (SC2) 26 162 Optimality conditions sufficient, semi-strong 2nd order 163 231 Optimality conditions sufficient, strong 2nd order 163 Optimality conditions sufficient, weak 2nd order 163 231 Optimality system 298 Oscillation 31 104 Parameter augmentation 222 Parameter identification 6 82 Parameter penalty 225 Partition of variables 315 Penalization see also "Function (penalty)" Penalization, exact 218 Penalization, exact, 232 Penalization, exact, augmented Lagrangian 218 223 233 Penalization, exact, Fletcher 231 263 Penalization, exact, Lagrangian 218 220 Penalization, exact, of the objective 218 Penalization, logarithmic 307 Piecewise line-search (PLS) see "Line-search" Pivoting 305 Point, basic 293 Point, basic, regular 304 Point, critical or stationary 26 27 52 82 161 297 Point, feasible or admissible 153 Point, interior 310 Point, optimal 97 Polak — Ribiere 73 Positive definite 26 27 67 82 83 86 Potential, Karrnarkar 389 Potential, logarithmic 307 325 Powell see "Correction" Preconditioning 34 Predicted decrease 49 80 Problem, ( 153 Problem, ( 169 Problem, constrained convex 26 153 Problem, convex 139 142 Problem, dual 98 295 296 Problem, least-squares 201 Problem, linear 292 Problem, primal 97 138 296 Problem, saddle-point 296 Problem, tangent quadratic, of an equality constrained problem 172 Problem, tangent quadratic, of an inequality constrained problem 205 Problem, tangent quadratic, of an unconstrained problem 167 Problem,quadratic 292 Projection onto a convex set 164 Proximity measure 311 369 Quasi-Newton or secant algorithm, quasi-Newton SQP 270 Quasi-Newton or secant algorithm, reduced, for equality constrained problems 279 Quasi-Newton or secant equation 54 267 Quasi-Newton or secant matrix 54 Quasi-Newton or secant method 56 78 86 Quasi-Newton or secant method, poorman 75 84 Quasi-Newton or secant update criterion 281 Reduced cost 303 Reduced Hessian of the Lagrangian 175 188 191 200 Regular stationary point for equality constrained problems 175 200 Relative distance 376 Row/column generation 145 Saddle-point 98 220 Safeguard 41 368 Scaling 35 Search see also "Line-search" Search, curvilinear 85 119 120 275 Secant see "Quasi-Newton" Self-duality 357 Separating hyperplane 97 103 Sequential quadratic programming, (SQP) algorithm 151
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