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Mangasarian O. — Nonlinear programming
Mangasarian O. — Nonlinear programming

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Название: Nonlinear programming

Автор: Mangasarian O.

Аннотация:

This reprint of the 1969 book of the same name is a concise, rigorous, yet accessible, account of the fundamentals of constrained optimization theory. Many problems arising in diverse fields such as machine learning, medicine, chemical engineering, structural design, and airline scheduling can be reduced to a constrained optimization problem. This book provides readers with the fundamentals needed to study and solve such problems. Beginning with a chapter on linear inequalities and theorems of the alternative, basics of convex sets and separation theorems are then derived based on these theorems. This is followed by a chapter on convex functions that includes theorems of the alternative for such functions. These results are used in obtaining the saddlepoint optimality conditions of nonlinear programming without differentiability assumptions.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1987

Количество страниц: 239

Добавлена в каталог: 21.08.2015

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$R^{n}$      6
Abadie, J.      97 99 100 205
Almgren, F.J.      62
Alternative, table of theorems of      34
Alternative, theorems of      27—37
Anderson, K.W.      3 205
Angle between two vectors      8
Arrow, K.J.      102 103 151 205
Avdeyeva, L.I.      212
Axiom of real number system      188
Ball, closed      183
Ball, open      182
Bartle, R.G.      200 205
Basis      178
Berge, C.      3 46 55 68 122 132 182 189 191 205
Bi-nonlinear function      149
Birkhoff, G.      187 205
Bohnenblust, H.F.      67 205
Boltyanskii, V.G.      72 163 210
Bolzano — Weierstrass theorem      189
Bounded function      196
Bounded set      188
Bounds, greatest lower and least upper      187
Bounds, lower and upper      187
Bracken, J.      205
Bram, J.      211
Brondsted, A.      122 205
Browder, F.E.      87 205
Buck, R.C.      182 188 205
Canon, M.      72 163 170 205
Caratheodory's theorem      43
Caratheodory, C.      43 205
Cauchy convergence      188
Cauchy — Schwarz inequality      7
Charnes, A.      206
Closed set      183
Closure of a set      183
Compact set      189
Concave function      56
Concave function, differentiable      83—91
Concave function, minimum of      73
Concave function, strictly      57
Concave function, strictly concave and differentiable      87—88
Concave function, strictly concave and twice differentiable      90—91
Concave function, twice differentiable      88—90
Concave functions, infimum of      61
Constraint qualification with convexity      78—81
Constraint qualification with differentiability      102—105 154—156 171—172
Constraint qualification, Arrow — Hurwicz — Uzawa      102
Constraint qualification, Arrow — Hurwicz — Uzawa, modified      172
Constraint qualification, Arrow — Hurwicz — Uzawa, weak      154
Constraint qualification, Karlin's      78
Constraint qualification, Kuhn — Tucker      102 171
Constraint qualification, reverse convex      103
Constraint qualification, reverse convex, weak      155 172
Constraint qualification, Slater's      78
Constraint qualification, Slater's, weak      155
Constraint qualification, strict      79
Constraint qualifications, relationships between      79 103 155
Continuous function      191—192
Convex and concave functions      54—68
Convex and concave functions, generalized      131—150
Convex and pseudoconvex functions, relation between      144 146
Convex combinations      41
Convex function      55
Convex function, continuity of      62
Convex function, differentiable      83—91
Convex function, differentiable at a point      83
Convex function, differentiable on a set      84
Convex function, strictly      56
Convex function, strictly convex and differentiable      87—88
Convex function, strictly convex and twice differentiable      90—91
Convex function, twice differentiable      88—90
Convex functions, fundamental theorems for      63—68
Convex functions, supremum of      61
Convex hull      44
Convex set      39
Convex sets      38—53
Convex sets, intersection of      41
Cooper, W.W.      206
Cottle, R.W.      122 206 210
Courant, R.      2 113 206
Cullum, C.      72 163 170 205
Daniel, J.W.      130
Dantzig, G.B.      15 18 113 122 206
Dennis, J.B.      113 206
Dieter, U.      122 206
Differentiable function, numerical      200
Differentiable function, vector      201
Distance between two points      7
Dorn, W.S.      117 122 124 125 206
Dual (maximization) problem (DP)      114 123
Dual linear problem (LDP)      127 130
Dual problem, unbounded      119 121 126 127
Dual variables      71
Duality      113—130 157—160 174—176
Duality in linear programming      126—130
Duality in nonlinear programming      114—123
Duality in quadratic programming      123—126
Duality theorem, Dorn's      124
Duality theorem, Dorn's strict converse      125
Duality theorem, Hanson — Huard strict converse      118
Duality theorem, linear programming      127
Duality theorem, strict converse      117 118 124 157 160 174 176
Duality theorem, weak      114 124
Duality theorem, Wolfe's      115
Duality with nonlinear equality constraints      174—176
Duffin, R.J.      32 206
Eisenberg, E.      122 206
Enthoven, A.C.      151 205
Equality constraints, linear      80 95
Equality constraints, nonlinear      75 112 161—176
Euclidean space $R^{n}$      6
Fan, K.      63 65 206
Farkas' theorem      16 31 37 53
Farkas' theorem, nonhomogeneous      32
Farkas, J.      16 31 206
Feasible directions, method of      111
Fenchel, W.      55 122 206
Fiacco, A.V.      206 207
Finite-intersection property      189
Fleming, W.H.      2 62 71 182 188 200 207
Fractional function, linear      149
Fractional function, nonlinear      148
Friedrichs, K.O.      113 207
Fritz John saddlepoint necessary optimality theorem      77
Fritz John saddlepoint problem (FJSP)      71
Fritz John stationary—point necessary optimality theorem      99 170
Fritz John stationary—point problem (FJP)      93
Fromovitz, S.      72 163 170 172 209
Function      11
Function, linear vector      12
Function, numerical      12
Function, vector      12
Gale's theorems      33
Gale, D.      16 33 35 113 127 177 179 207
Gamkrelidze, R.V.      72 163 210
Gass, S.      15 207
Ghouila Houri, A.      55 68 122 132 189 191 205
Glicksburg, I.      63 65 206
Gol'stein, S.      122 207
Gordan's Theorem      31
Gordan's theorem, generalized to convex functions      65
Gordan, P.      31 207
Gradient of a function      201
Graves, R.L.      207
Hadley, G.F.      15 207
Halfspace      40
Halkin, H.      72 163 170 207
Hall, D.W.      3 205
Halmos, P.R.      177 207
Hamilton, N.T.      3 207
Hanson, M.A.      113 117 118 122 137 207 208 210
Heine — Borel theorem      189
Hessian matrix      202
Hestenes, M.R.      200 208
Hilbert, D.      113 206
Hoffman, A.J.      63 65 206
Hu, T.C.      208
Huard, P.      115 117 118 122 208
Hurwicz, L.      102 103 122 205 208
Implicit function theorem      204
in the presence of equality constraints      80
Inequalities, linear      16—37
Infimum      187 195
Infimum of a numerical function      196
Interior of a set      183
Interior point      182
Jacob, J.-P.      122 208
Jacobian matrix      202
Jensen, J.L.W.V.      61 208
John, F.      77 99 208
Karamardian, S.      87 117 122 137 139 208
Karlin, S.      67 77—79 205 208
Koopmans, T.J.      208
Kowalik, J.      208
Krelle, W.      209
Kretschmar, K.S.      122 208
Kuhn — Tucker saddlepoint necessary optimality theorem      79
Kuhn — Tucker saddlepoint problem (KTSP)      71
Kuhn — Tucker stationary—point necessary optimality criteria      105 111 112 156 173
Kuhn — Tucker stationary—point problem (KTP)      94
Kuhn — Tucker sufficient optimality theorem      94 153 162
Kuhn, H.W.      79 94 102 105 113 127 207—209
Kunzi, H.P.      209
Lagrange multipliers      71
Lagrangian function      71
Landin, J.      3 207
Larsen, A.      122 209
Lax, P.D.      113 209
Levinson, N.      122 209
Levitin, E.S.      209
LIMIT      186
Limit point      186
Line      38
Line, segments      39
linear combination      6
Linear combination, nonnegative      7
Linear dependence      6
Linear inequalities      16—37
Linear minimization problem (LMP)      127 130
Linear programming: duality      126—130
Linear systems, existence theorems      21—26
Linearization Lemma      97 153 163
Local minimization problem (LMP)      70 93
MacLane, S.      187 205
Mangasarian, O.L.      72 117 118 122 131 140 151 157 163 170 172 209
Mapping      11
Martos, B.      132 137 151 209
Matrices      8—11
Matrix, nonvacuous      11
Matrix, submatrices of      10
Maximum      195
Maximum of a numerical function      197
Maximum principle      see "Minimum principle"
Maximum, existence of      198
McCormick, G.P.      205 207 209
Mean-value theorem      204
Metric space      7
Minimization problem (MP)      70 93
Minimization problem (MP), local      70 93
Minimization problem (MP), solution set      72
Minimization problem (MP), uniqueness      73
Minimum      195
Minimum of a numerical function      197
Minimum of a numerical function, existence of      198
Minimum principle      72 162—170 141—142
Minty, G.J.      87 209
Mishchenko, E.F.      72 163 210
Mond, B.      113 122 208—210
Moreau, J.-J.      122 210
Motzkin's theorem      28
Motzkin, T.S.      28 210
Negative definite matrix      91 181
Negative semidefinite matrix      89 90 181
Neustadt, L.W.      163 207
Nikaido, H.      131 132 210
Nonlinear programming problem      1—3 14—15
Nonsingular matrix      181
Norm of a vector      7
Notation      13—15
Numerical function      12
Oettli, W.      209
Open set      183
Opial, Z.      87 210
Optimality      18—21
Optimality criteria with differentiability      92—112 151—176
Optimality criteria, necessary      97—112 153—157 162—174
Optimality criteria, necessary saddlepoint      76—82
Optimality criteria, saddlepoint      69—82
Optimality criteria, sufficient      94—97 151—153 162
Optimality criteria, sufficient saddlepoint      74—76
Osborne, M.R.      208
Partial derivatives of a numerical function      200—201
Partial derivatives of a vector function      201—202
Plane      40
Point of closure      182
Polak, E.      72 163 170 205 209
Poly tope      41
Polyak, B.T.      209
Polyhedron      41
Ponstein, J.      117 118 122 209 210
Pontryagin, L.S.      72 163 210
Positive definite matrix      91 181
Positive semidefinite matrix      89 90 181
Primal minimization problem (MP)      114 122
Primal problem, no minimum      121 126 127
Product of a convex set with a real number      45
Pseudoconcave function      141
Pseudoconvex and strictly quasi-convex functions, relation between      143 146
Pseudoconvex function      140—141
Quadratic dual problem (QDP)      124
Quadratic minimization problem (QMP)      123
Quadratic programming      123—126
Quasiconcave function      132
Quasiconcave function, differentiable      134
Quasiconcave function, strictly      137
Quasiconvex and strictly quasiconvex functions, relation between      139 146
Quasiconvex function      131
Quasiconvex function, differentiable      134
Quasiconvex function, strictly      137
Rank of a matrix, column      179
Rank of a matrix, row      179
Relatively closed set      183
Relatively open set      183
Rissanen, J.      122 210
Ritter, K.      122 210
Rockafellar, R.T.      87 122 210
Rosen, J.B.      211
Rossi, P.      122 208
Rubinstein, G.S.      122 211
Rudin, W.      182 188 200 211
Saaty, T.L.      211
Saddlepoint problem, Fritz John (FJSP)      71
Saddlepoint problem, Kuhn — Tucker (KTSP)      71
Scalar product      7
Semicontinuous function, lower      192—193
Semicontinuous function, upper      193
Separating plane      46
Separation theorem      49
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