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| Mangasarian O. — Nonlinear programming |
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| Предметный указатель |
 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 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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