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| Magaril-Il'yaev G.G., Tikhomirov V.M. — Convex Analysis: Theory and Applications |
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| Предметный указатель |
Accumulation point 28
Admissible point (in an extremal problem) 70
Affine combination (of vectors) 31
Affine hull (of a set) 31
Affine independent points 31
Affine manifold 13 26 31
Affine subspace 13 26 31 80
Algebraic sum 30
Analog of Fermat's theorem 49
Annihilator 42 81
Banach — Alaoglu theorem 153
Barycentric coordinates 32
Basis 25
Bernstein's inequality 138
Biannihilator 42 81
Biconjugate cone 81
Bipolar 81
Blaschke's compactness theorem 68
Brunn — Minkowski inequality 67
Brunn — Minkowski symmetrization 67
Caratheodory's theorem 8 62
Cauchy theorem on rigidity 58
Cauchy — Steiner lemma 60
Cauchy's formula 65
Clean-up theorem 156
Clean-up theorem for cones 62
Closed set 4 27
Closure 28
Compact set 28
Complementary slackness condition 71
Cone 13 30
Conic combination 31
Conic hull (of a set) 31
Conjugate cone 42 81
Conjugate function 15 39
Conjugation operator 39
Conjugation operator for cones 42
Constraint 70
Continuous function 28
Convex analysis 1
Convex closure 7
Convex combination 31
Convex cone 13 31 80
Convex curve 1
Convex function 1 34
Convex homogeneous function 13
Convex hull 7
Convex hull of a set 31
Convex hull of a union (of convex sets) 53
Convex hull of minimum (of convex functions) 54
Convex polyhedron 88
Convex Problem 19 69
Convex programming problem 19 70
Convex set 1 30 31 80
Convex surface 1
Convolution (of convex functions) 54
Directional derivative 36
Distance between vectors 27
Dual problem 21
Dual problem (to an extremal problem) 73
Dual space 26
Dubovitskii — Milyutin formula 17
Dubovitskii — Milyutin theorem 52
Epigraph 1 34
Euclidean norm 27
Extremal subset 154
Extreme point 154
Farkas Theorem 90
Fenchel — Moreau theorem 11 16 44
Fermat — Torricelli — Steiner problem 96
Finitely generated cone 88
First separability theorem 38
Fredholm theorem 92
Fuller, A.T. 111
Gabushin, V.N. 111
Gale theorem 90
Generalized Gabushin's inequality 110
Generalized Hoelder's inequality 93
Goddard problem 97
Gruenbaum — Hammer theorem 68
Hahn — Banach — Kantorovich theorem 161
Half-interval 29
Half-space 37
Helly's theorem 9 63
Hoelder's inequality 93
Homogeneous function 13
Hyperplane 26 37
Image of a function under a mapping 35 54
Image of a set under a mapping 30 53
Indicator function 42
Indicator function of a convex set 34
Indicator operator 42
Interior of a set 27
Interior point of a set 27
Intersection 30
Intersection of convex sets 53
interval 29
Inverse image of a function under a mapping 35 54
Inverse image of a set under a mapping 30 53
Jensen inequality 34
| Karush — Kuhn — Tucker theorem 20 70
Kelley's sum of convex functions 54
Kelley's sum of convex sets 53
Kolmogorov-type inequalities 99
Ky Fan theorem 90
Lagrange function 70 163
Lagrange multipliers 70 163
Lagrange principle 20
Landau — Arestov theorem 118
Least possible constant 100
Legendre transform 39
Legendre — Young — Fenchel transformation 15
Lemma on nontriviality of the annihilator 168
Lemma on the annihilator of the kernel of a regular operator 165
Levin's theorem 52 156
Line segment 27 29
Linear combination (of vectors) 30
Linear programming problem 74
Linear span (of a set) 31
Locally convex space 13
Lyapunov, A.A., theorem 155 169
Lyusternik theorem 165
Magaril-Il'yaev theorem 116 118
Magaril-Il'yaev, G.G. 111
Masur theorem 160
Maximum (of convex functions) 35 54
Method of central sections 75
Method of circumscribed ellipsoids 76
Michael's selection theorem 161
Minkowski — M. Krein — D. Milman theorem 7 154
Minkowski's function 36 43
Minkowski's operator 43
Minkowski's theorem 6 64 90
Monge — Kantorovich problem 158
Moreau — Rockafellar formula 17
Moreau — Rockafellar theorem 51
Neighborhood 27
Nonatomic measure 155
Open ball 27
Open set 4 27
Perturbation of problem 21 72
polar 41 81
Polyhedral cone 88
Problem without constraints 20 49 70
Proper function 34
Rademacher theorem 160
Radon theorem 9 62
Ray 27
Relative interior (of a set) 32
Relatively interior point (of a set) 32
Riesz, F., theorem 92
Riesz, M., identity 138
S-function (of the family of problems) 72
Second conjugate 15 40
Second separability theorem 37
Second separation theorem 4 83
Separable sets 37
Separation theorems 16 37
Set of constraints 70
Sharp constant 100
simplex 31
Simplex method 77
Slater's condition 70
Solution (to an extremal problem) 70
Standard basis 25
Standard perturbation 73
Stationarity condition 101
Steiner — Minkowski formula 66
Straight line 27
Strictly differentiable mapping 164
Strictly separable sets 37
Subdifferential 17 40 48
Subdifferential form of the clean-up theorem 52
Subgradient 17 48
Sublinear function 13 35
subspace 13 26 30 80
Sum of convex functions 54
Sum of convex sets 53
Summation (of convex functions) 35
Support function 18 36 41
Support operator 41
Sz.-Nagy, B. 106 107
Sz.-Nagy, B. theorem 108
Tangent vector 164
Theorem on convex duality 43
Theorem on duality of sets 83
Theorem on existence and structure of subdifferential 50
Theorem on isoperimetric property of circle 94
Theorem on polyhedral sets 69
Theorem on strict separation 4
Theorem, Lagrange principle for smooth-convex problems 164
Theorem: the main formulas of convex calculus 55
Value (of an extremal problem) 70
Vertex 31
Weak topology 82
Weierstrass theorem 29
Weyl theorem 88
Young — Fenchel transform 39
Young's inequality 94
Zero-convex set 13 80
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