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Cambridge University Press 978-0-521-85005-6 - Convex Functions: Constructions, Characterizations and Counterexamples Jonathan M. Borwein and Jon D. Vanderwerff Index More information

Index

absolutely continuous, 79 approximate selection theorem, 112 absorbing set, 142, 142 approximately convex set, 193 in barreled spaces, 467 Archimedean solids, 462 abstract linear program, 108, 109 argmin, 285 unstable, 465 arithmetic-geometric mean inequality, 30 active set, 95 Aronszajn null, 197, 199, 201 adjoint, 46 Asplund averaging theorem, 221, 227 affine combination, 66 Asplund cavern, 474 affine function, 21 Asplund space, 199, 206, 223, 318, 318, 319, 320 affine hull, 66,69 and renorming, 220 affine mapping, 21 characterization, 281, 319, 321 characterization, 22 dual has RNP, 324 affine set, 66,69 scattered C(K), 220 Alaoglu’s theorem, 130 separable, 199, 215 Alexandrov’s theorem, 86 weakly compactly generated, 220 failure in infinite dimensions, 204 attainment, in Fenchel problems, 177 special cases in infinite dimensions, 475 α Attouch–Wets convergence, 289 -cone meager set, 197 Bicontinuity theorem, 290 (ANA) operator, see maximal monotone operator, of functions, 289 type (ANA) versus Mosco convergence, 295 analytic center, 182 versus pointwise convergence, 292 angle small set, 197, 199 versus slice convergence, 295 anti-proximinal, 327 versus uniform convergence on bounded sets, Applications in exercises 297 Blaschke–Santalo theorem, 32 Aumann convexity theorem, 14, 72 coupon collection problem, 180 Csiszár entropy problems, 366 differential inclusions, 123 direct and indirect utility, 56 Baire category theorem, 413, 467 divergence estimates, 58 Baire space, 460 doubly stochastic matrices, 93 balanced set, 18 experimental design duality, 58 Banach lattice, 329–337 Fisher information, 365 renorms, 334 gamma function, 32 Banach limits, 266 hidden convexity, 55, 56 Banach space, 22 location problem, 59 Banach’s fixed point theorem, 167 Lyapunov, duality, 124 barreled space, 460, 467 Lyapunov functions, duality of, 124 basic separation theorem, 29, 132 maximum entropy, 57 β-differentiability, 150, 379 monotonicity and elliptic PDE’s, 437 Šmulian’s theorem for, 159 multiplicative potential and penalty functions, Šmulian’s theorem for conjugate functions, 160 355 of norms, 227 NMR entropy, 63 BFGS update, 353 risk function duality, 185 biconjugate, see Fenchel biconjugate symbolic convex analysis, 63 bilinear map, 44 approximate fixed point, 167 as a second-order derivative, 44, 201 approximate mean value theorem, 166 symmetric, 88

© in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-85005-6 - Convex Functions: Constructions, Characterizations and Counterexamples Jonathan M. Borwein and Jon D. Vanderwerff Index More information

Index 509

bipolar, 178 polyhedron, 95 bipolar cone, see cone, bipolar range of multifunction, 114 bipolar theorem, 209 unit ball, 461 Bishop–Phelps cone, 182 compact separation theorem, 136 Bishop–Phelps theorem, 163, 169 complementary slackness, in cone programming, Blaschke–Santalo inequality, 33 110 Bohr–Mollerup theorem, 32 complemented subspace, 464 Boltzmann–Shannon entropy, 468 complete, 467 Borel measurable, 79 composing, USC multifunctions, 114 Borel measure, 195 composition, convex functions, 75 Borel sets, 305 concave bornology, 149, 187 conjugate, 183 Borwein’s variational principle, 170 concave function, 7, 40, 126 Borwein–Preiss smooth variational principle, 473 cone, 21 Bose–Einstein entropy, 364 bipolar, 67, 68, 108, 178, 178 boundary, properties, 462 ∗ finitely generated, 94 bounded weak -topology, 447 nonnegative, 468 (BR) operator, see maximal monotone operator, normal, see normal cone type (BR) pointed, 68,96 Brachistochrone problem, 10 characterization, 68 Bregman distance, 60, 358, 360 polar, 67, 67, 178, 183 Bregman projection, 361, 364 calculus, 178 Brøndsted–Rockafellar property, see maximal negative, 67, 178 monotone operator, type (BR) Brøndsted–Rockafellar theorem, 121, 162 positive, 67, 178 Brun–Minkowski inequality, 352 sum of closed cones, 183 program, 108 sum not closed, 183 Calderón norm, 316, 317 sums, 469 Carathéodory’s theorem, 71 tangent, see tangent cone category cone convex, 75 first, see first category cone quasiconvex, 335 second, see second category constraint, linear, 48, 107, 177, 182 Cauchy–Schwarz inequality, 22, 49 constraint qualification (CQ), 119, 405, 406 ˇ Cebyšev set, 8, 117, 188 constructible, 356, 356–357 convexity of, 8, 189, 190, 474 continuity in reflexive spaces, 188 absolute, 79 Cellina approximate selection theorem, 112, 281 ˇ and USC, 114 Chebyshev set, see Cebyšev set generic, 413 choice function, 336 in infinite dimensions, 460 Ck -smooth function, 222 of convex functions, 461 approximations, 230 ∞ of extensions, 116 C -smooth function, 222 of selections, 112 Clarke directional derivative, 82, 123 control theory, 472 Clarke subdifferential, 82 convex, quasi, 414 cusco property, 123 closed convex body, 33 images, 114 convex calculus theorem, see subdifferential, sum range, 461 rule subspace, 461 convex cone, base, 75 closed function, 2, 20, 127, 138 convex feasibility problem, 362 and lower-semicontinuity, 127 convex function, 18, 21, 126 and weak-lower-semicontinuity, 138 and monotone gradients, 37, 160 closure, of function, 21, 127, 127, 138 bounded above everywhere, 30 Cobb–Douglas function, 57 bounded on β-sets, 379 codimension, countable, 467 extensions of, 398 coercive function, 24, 174 bounded on bounded sets, 137, 379, 382, 384 conjugate of, 174, 175 conjugate supercoercive, 175 minimization on reflexive space, 183 extensions of, 399 versus cofinite, 185 bounded on weakly compact sets, 379, cofinite function, 118, 174, 361 383, 384 versus supercoercive, 118, 185, 361, 384 composition, 75 compact conjugate, see Fenchel conjugate images, 114 continuity of, 4, 25, 128, 129, 461 in infinite dimensions, 461 continuous biconjugate, 386

510 Index

convex function (Cont.) minimal, 279, 283 continuous extensions, 386, 392–399 failure, 392 continuously differentiable, 35 (D) operator, see maximal monotone operator, difference of, see DC function type (D) epigraph, 139 Darboux property, 72 essentially smooth, see essentially smooth Davis’ theorem, 101, 103, 472 function DC function, 104, 146 essentially strictly, see essentially strictly approximation of Lipschitz functions, 218 convex function failure, 147 finitely extendable, 398 k-th largest eigenvalue, 11, 104 Fréchet differentiable, see Fréchet vector-valued, 104 differentiability DC operators, 317 Gâteaux differentiable, see Gâteaux dense differentiability hyperplane, 467 Legendre, see Legendre function range, 469 Lipschitz, 51, 144, 173 subspace, 469 extensions of, 53, 399 dense type, 439, 453 on bounded sets, 128, 137, 141 dentable, 321 local and global minima, 4, 26, 130 denting point, 326 local Lipschitz property, 4, 25, 128 derivative locally uniformly, see locally uniformly convex, Dini, see Dini derivative function directional, see directional derivative midpoint, see midpoint convex function Fréchet, see Fréchet differentiability on real line, 20, 36 Gâteaux, see Gâteaux differentiability recognizing, 36, 37, 39, 91, 141, 160, 202 determinant, order preservation, 103 stability properties, 3, 23, 139 diametral set, 167 strictly, see strictly convex, function difference convex function, see DC function strongly rotund, see strongly rotund function difference quotient, 84 totally, see totally convex function second-order, 84 unbounded on bounded set, 381 differentiability uniformly, see uniformly convex, function generic, 413 uniformly smooth, see uniformly smooth, of spectral functions, 102 function differential inclusion, 123 weak lower-semicontinuity, 132 Dini derivative, 79 convex functions, matrix, 105 directional derivative, 6, 26, 131, 147 convex hull of function, 21, 128 Lipschitz property, 131 of set, 21, 126, 464, 465 lower, 166 convex mean value theorem, 121 sublinear property, 27, 131 convex program, 177 disjoint operator ranges, 469 ordinary, 48 distance function, 8, 22, 127, 180 convex series closed, 142 attainment, 461 convex set, 18, 126 convexity of, 23 characterization in Hilbert space, 190 on Hilbert space, 189, 190 stability properties, 139 distortable, arbitrarily, 193 symmetric, 18 distortion, 189 convexity, 430 of a norm, 192 core, 5, 25, 29, 34, 129, 129, 131, 141 divergence, Kullback–Leibler, 13 and absorbing sets, 142, 467 divergence estimates, 75 in barreled spaces, 467 domain versus interior, 25, 142 of a function, 5, 18 countable of a monotone operator, 118 basis, 467 of subdifferential codimension, 467 not convex, 141 countable set, 195 polyhedral, 95 Csiszár divergence, 365 domain regularizable, 412 cusco, 111, 279 doubly stochastic, 93 ∗ and Asplund spaces, 281 DP -property, 383, 383 and maximal monotone operators, 280 dual linear program, 107 Clarke subdifferential, 123 dual norm, 129 fixed points on compact convex, 111, 281 characterization, 210 ∗ generically single-valued, 283 implied by weak -Kadec property, 224 maximal monotone multifunction, 123 dual space, 129

Index 511

dual value, in LP and SDP, 107 and coercivity, 347 duality duality with essential smoothness, 344 gap examples, 353 Duffin’s, 50 in the classical sense, 338 LP, 107 compatibility in Euclidean space, 345 SDP, 107 outside of Euclidean space, 346 weak, 177 on reflexive spaces, 345 Fenchel, 48, 98 spaces without, 373 duality mapping, 133 spectral functions, 102 Dunford–Pettis property, 383, 385, 386, 388 versus strict convexity, 341, 342, 344, 347, 374 Dunford–Pettis theorem, 302 Euclidean space, 18, 22, 460 excess functional, 289 experimental design, 58 (ED) operator, see maximal monotone operator, exposed point, 92, 211 type (ED) duality with differentiability, 211 eigenvalue of function, 232, 235 k-th largest, 11, 104 characterization, 232 mapping, 314 duality with Gâteaux differentiability, 234, eigenvalue problem, 438 240 eigenvalues versus extreme point, 235 functions of, 101 versus strongly exposed, 234 optimization of, 472 strongly, see strongly exposed point Ekeland’s variational principle, 161, 168 versus strongly exposed, 92 equivalence with completeness, 166 extension, continuous, 116 in Euclidean space, 117 extreme direction, 335 entropy extreme point, 92, 211 Boltzmann–Shannon, 61 of polyhedron, 95 Bose–Einstein, 364 versus exposed point, 92 Burg, 61 versus support point, 224 nonattainment, 311 energy, 61 Fermi–Dirac, 61, 364 Fan inequality, 99 maximum, 179, 182 Fan minimax theorem, 55 nonattainment, 310 Fan theorem, 99 relative, 58 Fan–Kakutani fixed point theorem, 111, 281 Tallis, 480 Farkas’ lemma, 70 epigraph, 2, 20, 127 farthest point function, 167 as multifunction graph, 114 Favard inequality, 49 polyhedral, 94 feasible set, 49 epigraphical convergence Fenchel biconjugate, 44, 96, 101, 103, 171, 178, and minimizing sequences, 293 179 relations among, 291, 295, 296 Fenchel biconjugation theorem, 65 relations among in finite dimensions, 296 Fenchel conjugate, 44, 171, 179 summary of relations among, 291 and eigenvalues, 100 versus pointwise convergence, 292 bounded on bounded sets, 384, 385 ε-net, 318 bounded on weakly compact sets, 385 ε-subdifferential, 121, 137 differentiable, 187 nonempty, 137 examples, 45 of conjugate function, 160 of coercive function, 174, 175 of norms, 222 of composition, 76 equilibrium, 473 of exponential, 179, 182 essentially smooth function, 338, 362 of infimal convolution, 176 barrier for open convex cone, 351 of Lipschitz convex, 51, 173 barrier for open convex set, 348, 370 self conjugate characterization, 179 characterizations, 344 transformations, 46 duality with essential strict convexity, 344 Fenchel duality, 99, 177 examples, 353 and LP, 108 Fréchet, 347 generalized, 99 in the classical sense, 338 linear constraints, 48, 97, 182 compatibility in Euclidean space, 345 polyhedral, 97, 98 outside of Euclidean space, 346 symmetric, 183 log barriers, 100 Fenchel duality theorem, 6, 46, 177 spectral functions, 102 Fenchel problem, 177 versus Gâteaux differentiability, 341 Fenchel–Young inequality, 44, 101, 171 essentially strictly convex function, 339, 362 Fermat point, 59

512 Index

Fermat–Weber problem, 59 Lyapunov, 124 Fermi–Dirac entropy, 364 nowhere differentiable, 8 Fillmore–Williams theorem, 104 simple, 302 finitely generated strongly rotund, see strongly rotund cone, 94–96 function function, 94–98 fundamental theorem of calculus, 79 set, 94–98 finitely representable, 143, 225 Gâteaux bornology, 149 first category, 195 Gâteaux differentiability, 34, 37, 149, 150, 198, Fisher information, 365 362 Fisher information function, 365 and continuous selections, 277 Fitzpatrick function, 119 and directional derivatives, 41 fixed point, 167, 474 and interior of domains, 157 Fourier identification, 469 and subdifferentials, 150 (FP) operator, see maximal monotone operator, and unique subgradients, 4, 34, 151, 158, 346, type (FP) 462 Fréchet characterizations of, 34, 151, 152, 154 smooth space, 166 continuity of, 35, 43, 154 subderivative, 165 duality with exposed points, 234, 240 subdifferentiable, 165 duality with strict convexity, 240, 250, subdifferential, 165 341 monotonicity of, 414–415 generic, 198 Fréchet bornology, 149 nongeneric, 199, 284 Fréchet differentiability, 34, 37, 149, 150, 198 of conjugate functions, 160, 187 almost everywhere, 77, 78 of norms, 150, 210,211 and continuous selections, 277 and unique support functionals, 212 and unique subgradients, 4, 34 duality with exposed points, 211 characterizations of, 34, 151, 153 duality with strict convexity, 192 continuity of, 35, 43, 154 nowhere, 205 dual norm, 221 versus Fréchet differentiability, 227 duality with local uniform convexity, 240, 251 second-order, see second-order derivative duality with perturbed minimization, 237 sets of nondifferentiability points, 201 duality with strongly exposed points, 234, 240, Šmulian’s theorem for, 154 250 symmetric formulation, 152 duality with Tikhonov well-posed, 235 uniform, see uniformly Gâteaux differentiable generic, 77, 152, 199, 215 versus continuity, 36, 150 conjugate functions and RNP, 321, 323 versus Fréchet differentiability, 4, 34, 36, 41, on Asplund spaces, 319 381, 382, 384 implies continuity, 41 versus weak Hadamard differentiability, 383, of conjugate functions, 160, 187 385 and biconjugates, 178 Gâteaux differentiability space, 161, 198 and convexity of original, 179 -function, 32 and minimization principles, 323 gauge function, 22, 127 and reflexivity, 224, 228, 250, 251 and lower semicontinuity, 144 of norms, 150, 210, 215, 220 and polar function, 58 and separable duals, 215 implicit differentiation, 154 characerization, 210 properties of, 30, 144 duality, 225 gauge of uniform convexity, 266 duality with strongly exposed points, 211 Gelfand–Phillips space, 383 nowhere, 205 generated cuscos, 116 on Asplund spaces, 319 generic versus Gâteaux differentiability, 227 continuity, 413 second-order, see second-order derivative differentiability, 413 Šmulian’s theorem for, 153 single-valued, 413 support functions, 317, 318 generic set, 195 symmetric formulation, 151 Goldstine’s theorem, 140 uniform, see uniformly Fréchet differentiable good asymptotical behavior, 340 versus Gâteaux differentiability, 4, 34, 41, 381, Gordan’s theorem, 70 382, 384 Gossez operator, 446 versus weak Hadamard differentiability, 379, Grüss–Barnes inequality, 49 381, 384, 385 graph Fubini’s theorem, 79 minimal, 116 function of a monotone operator, 278 Kadec property, see Kadec property, function of a multifunction, 111

Index 513

Grothendieck property, 383 inf-compact, 340 Grothendieck space, 383, 390, 391, 396, 399 infimal convolution, 51, 53, 175 continuous biconjugate function, 386 and approximation, 176, 184 extensions of convex functions, 396 and conjugation, 52 and uniform convergence, 52 as a kernel average, 180 Haar null, 196, 200 basic properties, 51 Hahn–Banach extension property, 147 conjugate of, 176 Hahn–Banach extension theorem, 28, 130 injective Banach space, 148 Hahn–Katˇetov–Dowker sandwich theorem, 116 inner product, 143 Hamel basis, 128, 467 inverse, image, 97 Hardy et al. inequality, 100 isotone, 75 Harmonic-arithmetic log-concavity theorem, 33 Hausdorff distance, 289 Hessian, 38 James space, 458 and convexity, 39, 202 James’ theorem, 143 higher-order derivative, 221 James–Enflo theorem, 218 of norms on Lp-spaces, 222 Jensen’s inequality, 29 Hilbert cube, 141 JN-sequence, 382 Hilbert space, 144, 464 jointly convex, 60,61 characterizations, 481 Josefson–Nissenzweig theorem, 382 isomorphic characterizations, 273 lattice characterizations, 332 Kadec property, 223 Hilbert–Schmidt operators, 313 function, 306 Hölder exposed point, 268, 273 Kadec property, weak see Kadec–Klee property ∗ duality, 270 weak , 191, 223 Hölder inequality, 49 and Asplund spaces, 223 Hölder smooth point, 269, 273 implies norm is dual, 224 hyperplane, dense, 467 of double dual, 224 hypersurface, 201 Kadec’ theorem, 214 Kadec–Klee property, 189, 190, 193 ∗ Kadec property, weak see Kadec property, ideal, 331 ∗ implicit function theorem for gauges, 154 weak improper function, 21 Kakutani–Fan fixed point theorem, 111, 281 improper polyhedral function, 98 kernel average, 180 inclusion control problem, 401 key theorem, 94, 96 incomplete normed space, 460 Kirchberger’s theorem, 72 inconsistent, 109 Kirszbraun–Valentine theorem, 282 indicator function, 3, 21, 127, 180 Klee cavern, 474 conjugate of, 179 Knuth’s inequality, 41 subdifferential of, 130 Krein–Milman property, 324 induced norm, 106 Krein–Milman theorem, 140 inequality Krein–Rutman theorem, 67 arithmetic-geometric mean, 30 Kullback–Leibler entropy, 363 Blaschke–Santalo, 33 Kuratowski–Painlevé convergence, 285 Brun–Minkowski, 352 and minimizing sequences, 287 Cauchy–Schwarz, 22, 49 of functions, 286 Fan’s, 99 not preserved under conjugation, 296 versus pointwise convergence, 292 Favard, 49 versus Wijsman convergence, 295 Fenchel–Young, 44, 171 Fitzpatrick, 435 Grüss–Barnes, 49 Lagrangian Hardy et al., 100 duality Hölder, 49 linear programming, 107 Jensen’s, 29 Lagrangian dual, 49 Knuth’s, 41 Lambert W function, 75, 76 Lyapunov duality, 124 Laplacian, 438 polar, 58 lattice, ordering, 333 Rogers–Hölder, 352 lattice band, 331 strong Fitzpatrick, 451 lattice operations, 329 three-slope, 19 Lau–Konjagin theorem, 189 triangle, 22 Lebesgue integral space, 143 variational, 436 Lebesgue sequence space, 143 weak duality, 7, 47, 177 Lebesgue–Radon–Nikodým theorem, 304

514 Index

Legendre function, 339, 364 extensions of, 229 cofinite versus supercoercive, 361 on separable space, 214 duality failure outside reflexive spaces, 348 on WCG spaces, 220 duality in reflexive spaces, 344 preserved by sums, 225 examples, 353, 354 locally uniformly rotund, see locally uniformly Fréchet, 347 convex in the classical sense, 338 location problem, 59 compatibility in Euclidean space, 345 Loewner convex functions, 105 outside of Euclidean space, 346 Loewner ordering, 61, 106 on general Banach space, 369 log, 100, 100, 182 on reflexive spaces, 345 log barrier, 100 spaces with/without, 373 log det, 100, 100–103, 353 spectral, 353 log-convex function, 40, 40,42 zone consistency of, 362, 368 lower level set, 20, 127 Legendre transform, 348, 375 bounded, 31, 46, 185 Legendre–Fenchel conjugate, see Fenchel, and coercivity, 24, 174 conjugate closed, 127 lexicographic order, 314 compact, 100 lim inf i→∞ Fi, 285 convex, 23, 139 lim supi→∞ Fi, 285 of conjugate function, 175 linear of differentiable function, 154 unbounded, 50 objective, 107 ∗ programming (LP) weak -closed, 138, 210 abstract, 108, 109 weakly closed, 138 and Fenchel duality, 108 lower-semicontinuous primal problem, 107 and USC, 114 linear functional generic continuity, 413 continuity of, 461 in infinite-dimensions, 460 discontinuous, 142, 144, 460, 461 sandwich theorem, 116 non-norm-attaining, 468 lower-semicontinuous function, 2, 20, 98, 127 norm-attaining, 133, 143 equivalent to closed, 127 τ dense in dual, 163, 169 -lsc, 130 supporting, 133, 468 weakly-lsc when convex, 132 dense, 169 LSC multifunction, see multifunction, LSC unique and differentiability, 212 Lyapunov convexity theorem, 15 Lipschitz, eigenvalues, 104 Lyapunov function, 124 Lipschitz constant, 19, 173 Lipschitz exposed point mathematical economics, 473 of a function, 268, 270 matrix of a set, 268 analysis, 100 of functions, 273 completion, 353 Lipschitz extension, 53 optimization, 107 Lipschitz function, 19, 127 matrix norm, 106 Lipschitz smooth point, 204, 269, 273 max formula, 6, 28, 131 implied by weak second-order Taylor, 204 max function, subdifferential of, 145 nongeneric, 270 maximal monotone multifunction, see maximal locally bounded, 114, 116 monotone operator locally Lipschitz function, 19 maximal monotone operator, 118, 278 Fréchet differentiable almost everywhere, 78 acyclic, 418, 418–423 locally uniformly convex and cuscos, 280 dual norm, 215, 224 and subdifferentials, 118, 278, 413, 419, 429 C1-smooth approximations, 230 Asplund decomposition, 418 and smoothness, 213 coercive, 446 on separable dual, 214 versus surjective, 437 function, 239 continuity, 283 and renorming, 246 convexity of domain and range, 410–412, 452, and strongly exposed points, 239 458 at a point, 239, 272 cyclically, 429 duality with Fréchet differentiability, 240, dense type, see type (D) 251 irreducible, see acyclic is totally convex, 374 locally maximal monotone, see type (FP) stable under sums, 241 maximal monotone locally, see type (VFP) norm, 213, 220, 221, 246 on reflexive spaces, 433–439 duality, 225 relations between classes of, 459

Index 515

strongly locally maximal, 442, 447 and minimizing sequences, 288, 293, 294 sum theorem on reflexive space, 434 and optimal value problems, 311 type (ANA), 455, 455 and the Kadec property, 299 type (BR), 455 bicontinuity theorem, 288 type (D), 439, 442, 446, 482 stability under addition, 300 implies type (FP), 441 versus Attouch–Wets convergence, 295 type (ED), 439, 482 versus pointwise convergence, 292 type (FP), 410, 410, 441, 446 versus slice convergence, 296 type (NI), 439, 440, 445, 450–459, versus Wijsman convergence, 296, 299 482 multifunction, 111, 276 characterization, 453 closed, 111 sum theorem, 450 versus USC, 114 type (VFP), 410, 441, 442 continuous, 111 type (WD), 453 cyclically monotone, 415 type negative infimum, see type (NI) domain of, 276 Mazur’s theorem, 198 fixed point, 111 meager set, 195 isc, 111 mean value theorem, 121 locally bounded, 122 approximate, 166 locally bounded at a point, 279 measures, tight, 207 LSC, 111, 113–116 method of cyclic projections, 363, 363 monotone, see monotone operator N-monotone Michael selection theorem, 113, 281 415 midpoint convex function, 31, 139 osc, 111 measurable, 31, 139 quasimonotone, 414 Mignot’s theorem, 412 range of, 111, 112, 114, 281 Milman–Pettis theorem, 217 USC, 111, 276 minimal, graph, 116 minimax theorem, 182 nearest point, 53 minimization principles, 273 in infinite dimensions, 464 minimizer, subdifferential zeros, 131 in polyhedron, 182 Minkowski, 472 selection, 114, 413 functional, see gauge function to p-sphere, 54 theorem, 95 to an ellipse, 54 minorant, affine, 97 nearest point mapping, 54, 188 modulus of convexity discontinuous, 54 of functions, 244 empty, 191 at a point, 272 nowhere differentiable, 204 duality, 252 necessary optimality condition power type duality, 253 Pshenichnii–Rockafellar, 147 versus gauge of uniform convexity, 266 negligible set, 197 of norms, 215, 258 NI Banach space, 450 duality, 217, 218 (NI) operator, see maximal monotone operator, power type, 2, 218, 226, 260 type (NI) modulus of smoothness nonatomic, 72 of functions, 252 nonexpansive at a point, 269 mapping, 282 duality, 252, 253 extension, 282 power type, 255, 256 nonnegative cone, 468 of norms, 216 nonreflexive Banach space, 184, 269 duality, 217, 218 DC function characterization, 146 power type, 218, 255, 256 Haar null sets, 200, 206 monotone multifunction, see monotone operator lack of nearest points, 191 monotone net property, 77 Legendre functions, 348 monotone operator, 118, 276 norm, 22 differentable, 412 dual, see dual norm extension formulas, 407–408 entrywise, 107 locally bounded, 279, 408–410 induced, 106 maximal, see maximal monotone operator subgradients of, 145 upper-semicontinuous, 276 submultiplicative, 106 monotone set, 278 norm topology, 461 monotonically related, 118 normal integrands, 305, 306 Moreau–Rockafellar dual theorem, 175, 463 normal cone, 26, 67, 144, 147, 178 Moreau–Rockafellar theorem, 46, 118, 175 and polarity, 67 Mosco convergence, 287 and subgradients, 182

516 Index

normal cone (Cont.) prepolar, 209 and tangent cone, 68 primal to intersection, 182 linear program, 107 normal mapping, 436 semidefinite program, 109 normed linear space, 22, 467 principle of uniform boundedness, 16 nowhere dense, 195 proper function, 2, 21,95 null set, 195 property Aronszajn, see Aronszajn null Krein–Milman, 324 Haar, see Haar null Radon–Nikodým, see Radon–Nikodým property proximal set, 188 objective function, 48 in reflexive spaces, 188 linear, 107 Pshenichnii–Rockafellar conditions, 73, 147 open map, 30 open mapping theorem, 30, 142 optimal value, 97 quadratic form, conjugate, 106 dual, 177 quadratic programming, Boolean, 110 in LP and SDP, 107 quasi-complement, 466 primal, 177 quasi-Newton method, 353 optimization quasiconvex, 414, 430 linear, 107 with respect to a cone, 335 quasimonotone multifunction, see multifunction, matrix, 107 quasimonotone subdifferential in, 130 order preservation Rademacher’s theorem, 78 of determinant, 103 in infinite-dimensions, 199 order-complete, 147 radius function, 167 ordering, lattice, 333 Radon measure, 195 orthogonal Radon–Nikodým property, 321, 324, 325 invariance, 103 and Krein–Milman property, 324 similarity transformation, 103 sets with, 321, 321 osc multifunction, see multifunction, osc duality with differentiability, 321 minimization principles, 324 strongly exposed points, 322, 324 p-norm, 49, 143 sets without, 327 paired Banach spaces, 313 spaces with, 321 paracompact, 281 characterization, 325 Pareto efficient point, 336 dual to Asplund, 324 partial order, directed, 335 minimization principles, 323 partition of unity, 111–115 no anti-proximinal pairs, 327 permutation, matrix, 104 and Rademacher’s theorem, 199 Pinsker inequalities, 13 smooth bump functions, 328 Pisier’s theorem, 218 smooth variational principles, 328 ∗ Pitt’s theorem, 325 weak -Asplund dual, 323 polar cone, see cone, polar spaces without, 327 polar of a set, 33, 209 range, closed, 461 polyhedral range of multifunction, 114 algebra, 97–98 ray, 462 calculus, 98 rearrangement, 313 cone, 95, 99, 108 invariant, 313 convex analysis, 94 recession function, 50 Fenchel duality, 97 reflexive Banach space, 143, 288 function, 94–99 and nearest points, 189 problem, 107 Cebyševˇ sets, 188, 190, 193 polyhedron, 73, 94–99, 147 coercivity and minimization, 183 compact, 95 dual norms, 192, 221, 224, 225 nearest point in, 182 essential strict convexity and smoothness polyhedral set, 94 duality, 344 tangent cone to, 98 essentially strictly convex functions, 345, 347 polytope, 94–96 Fréchet differentiable conjugate, 224, 228, 250, porous set, 197 251 σ -, 197 Haar null sets, 206 positive definite, strongly, 231 Legendre functions, 344, 345, 358, 362, 368 positive operator, 312 local uniform convexity/ Fréthet positively homogeneous function, 22 differentiability duality, 251

Index 517

nonreflexive, see nonreflexive Banach space versus Taylor expansion, 89 norm-attaining functionals, 143 selection, 111, 277, 281 proximal sets, 188 continuous, 277 strict convexity/Gâteaux differentiability continuous and differentiability, 277 duality, 240, 250, 341 self-adjoint operator, 312 stronly exposed point/Fréchet differentiability self-dual cone, 67, 101, 109 duality, 240 semidefinite uniformly convex norm, 217 cone, 67, 100, 103, 107 uniformly smooth norm, 217 program (SDP), 107 weak compactness, 143 semidifferentiable, 300 regularity, condition, 97 subdifferential, 300 relative interior, 66,69 twice, 300 and Fenchel duality, 99 separable, 182 and separation, 67 and semicontinuity, 469 and subdifferentials, 67 Banach space, 468 in infinite dimensions, 462 separately convex, 60,61 nonempty, 4, 66, 462 separating family, 140 relaxation, semidefinite, 110 separation renorm, in Ekeland principle, 170 and bipolars, 68 renorms failure without interior, 139, 462 of Banach lattices, 334 in infinite dimensions, 462 of Banach spaces, 213–215, 218–222, 224, 225, separation theorem, 64, 67, 132, 133 227, 229, 246, 254, 258, 260, 264, 274 sequence of sets residual, 195 Kuratowski–Painlevé limit, 285 locally, 206 lower limit, 285, 292 resolvent, 412 upper limit, 285, 292 Riccati equation, 367 set-valued mapping, see multifunction Rockafellar function, 429 simple function, 302 Rockafellar’s theorem, 118, 278 simultaneous ordered spectral decomposition, Rogers–Hölder inequality, 352 100, 102 single-valued Sandwich theorem, 7, 133 generic, and maximal monotonicity, 413 Hahn–Katˇetov–Dowker, 116 singular value, 106 scattered, 220 skew operator, 88, 416 Schatten p space, 313 Fitzpatrick–Phelps, 449 Schur property, 4, 384, 384 Slater’s condition, 50, 108 dual spaces with, 386 slice convergence, 290 second category, 195 and minimizing sequences, 293 second-order derivative and renorming, 290 and convex functions, 39, 202 and the Kadec property, 298 and Taylor expansions, 84, 203 versus Attouch–Wets convergence, 295 equivalent properties, 84 versus Mosco convergence, 296 Fréchet, 38, 38, 201 versus Wijsman convergence, 298 and Taylor expansions, 84 smooth approximation, 230 generalized, 84, 84, 85, 86, 88, 90 on superreflexive spaces, 231 generalized and Taylor expansions, 203 smooth variational principle, 165 ∗ generalized weak , 202, 203 higher-order, 328 nowhere, 204 Šmulian’s theorem symmetry of, 206 for β-differentiability, 159 versus Gâteaux, 89 for conjugate functions, 160 ∗ weak , 201 for Fréchet differentiability, 153 Gâteaux, 38, 44, 90, 201 for Gâteaux differentiability, 154 and Taylor expansions, 84 for norms, 210, 211 generalized, 84, 84, 86, 88, 203 for uniform Fréchet differentiability, 155 ∗ generalized weak , 202 for uniform Gâteaux differentiability, 156 ∗ generalized weak and Taylor expansions, Sobolev space, 332, 437 203 space nowhere, 204 Asplund, see Asplund space symmetry of, 206 Baire, see Baire space versus Fréchet, 89 barreled, see barreled space ∗ weak , 201 Euclidean, see Euclidean space generalized Hilbert, see Hilbert space almost everywhere, 86 reflexive, see reflexive Banach space symmetry of, 38, 84, 206 superreflexive, see superreflexive Banach space

518 Index

space (Cont.) subdifferential, 3, 26, 130, 165 uniformly convex, see uniformly convex, space at boundary points, 43, 339, 340 uniformly smooth, see uniformly smooth, space at optimality, 6, 26, 130 ∗ ∗ weak -Asplund, see weak -Asplund space boundedness properties, 32, 137 weakly compactly generated, see weakly compact and convex, 29 compactly generated space continuity properties, 43 spectral convergence properties, 43 conjugacy, 101, 103 cusco property, 123 differentiability, 102 domain not convex, 141 function, 100 empty at a point, 131 convex, 101, 103 empty on dense set, 141 sequence, 313 failure of sum rule, 146 sequence space, 313 limits of gradients, 78 subgradients, 102, 104 local boundedness, 32, 122, 138, 340 spectral operators, Fréchet differentiability, 316 maximal monotonicity of, 118, 121, 278, spectral radius, 106, 231 413 square-root iteration, 103, 316 monotonicity of, 42, 160, 414 staunch, 208 near boundary points, 70 staunch set, 197 nonempty, 67, 131 Stegall’s variational principle, 324 on dense set, 162 Steiner problem, 59 not onto, 184 Straszewicz’s theorem, 92 of conjugate functions, 160 strict maximum, 232 of indicator function, 130 strict minimum, 232 of max function, 145 strictly convex of norm, 145 dual norm, 213 of polyhedral function, 99 and Gâteaux smooth approximations, 230 of support function, 148 and smoothness, 213 singleton versus differentiability, 34, 151, 158, on dual to WCG, 220 462 function, 18, 126, 238, 362 sum rule, 7, 48, 135 and exposed points, 239, 240 upper-semicontinuity, 276 and extreme points, 239, 240 ∗ weak -closed and convex, 145 at a point, 239 subgradient, 3, 26, 130 duality with Gâteaux differentiability, 240, and normal cone, 182 250, 341 existence of, 97 recognizing, 39, 202 of maximum eigenvalue, 145 stable under sums, 241 versus essential strict convexity, 341, 342, of norm, 145 344, 374 of polyhedral function, 95 log barriers, 100 unique, 462, 468 norm, 188, 212, 213, 221 subjets, 472 and extreme points, 212 sublevel set, see lower level set characterizations, 212 sublinear function, 22, 103 duality, 192 almost, 31 extensions of, 229 characterization, 22 preserved by sums, 225 subspace spaces without, 220 closed, 461 strong maximum, 232, 233 complemented, 464 strong minimum, 163, 232, 233 countable-codimensional, 467 strongly exposed point, 211 dense, 469 duality with differentiability, 211 substitution norm, 334 of function, 233 sum, of subspaces, 469 and coercivity, 236 sum rule for subdifferentials, see subdifferential, characterization, 233 sum rule duality with Fréchet differentiability, 234, sun, 194 240 supercoercive function, 118, 174, 339 versus exposed, 234 conjugate of, 175 versus exposed point, 92 on reflexive space, 183 strongly rotund function, 306 versus cofinite, 118, 185, 361, 384 and optimal value problems, 311 superreflexive Banach space, 143, 225, 274 convex integral functional, 309 and smooth approximation, 231 subadditive function, 22 characterization, 218, 264 subderivative, 165 renorming, 218, 221 subdifferentiable, 165 smooth variational principles, 328

Index 519

support function, 6, 22, 127, 179, 317, 318, Fan’s minimax, 55 326 Farkas’ lemma, 70 subdifferential of, 148 Fenchel biconjugation, 65 support point, 133, 468 Fenchel duality, 6, 46, 177 dense in boundary, 169 first-order condition, 6 proper, 468 Gauss–Lucas, 9 supporting hyperplane, 65, 133 Goldstine, 140 supporting hyperplane theorem, 65 Gordan’s, 70 supporting linear functional, see linear functional, Hahn–Banach extension, 28, 130 supporting Hahn–Katˇetov–Dowker sandwich, 116 surjective, linear map, 108 harmonic-arithmetic log-concavity, 33 symmetric Helly, 5 function, 100 Hessian and convexity, 39, 202 set, 18, 104 James, 143 symmetric operator, 416 James–Enflo, 218 Josefson–Nissenzweig, 382 tail operator, 456 Kadec, 214 tangency properties, 462 Kakutani–Fan fixed point theorem, 111, 281 tangent cone, 68 Kenderov, 413 to polyhedron, 98 key, 94 τβ -topology, 150, 187 Kirchberger’s, 72 -sequentially lsc, 187 Krein–Milman, 140, 336 properties of convex functions, 379 Krein–Rutman, 67 Taylor expansion Krein–Šmulian, 186 second-order Lau–Konjagin, 189 almost everywhere, 86 Lebesgue–Radon–Nikodým, 304 and differentiability, 203 Levy–Steinitz, 9 equivalent properties, 84 Lidskii, 313 implies first-order differentiability, 203 Lyapunov convexity, 15 nongeneric, 91 max formula, 6, 28, 131 strong, 83, 88, 202, 203 Mazur, 198 strong versus weak, 84, 202 Michael selection, 113, 281 versus differentiability of nonconvex, 89 Mignot’s, 412 versus mixed partials, 89 Milman converse, 92 weak, 83, 88, 202, 203, 204 Milman–Pettis, 217 third-order Minkowski, 8, 92 versus differentiability of convex, 89 Minty condition, 119 theorem monotone gradients and convexity, 37, 160 prime ideal, 337 Moreau–Rockafellar, 46, 118, 175 Alaoglu, 130 Moreau–Rockafellar dual, 175, 463 Alexandrov, 86 open mapping, 30, 142 approximate mean value, 166 Pisier’s, 218 Asplund averaging, 221, 227 Pitt, 325 Aumann convexity, 14, 15, 72 Rademacher, 78, 199 Banach’s fixed point, 167 Radon, 5 Banach–Dieudonné, 186 Banach–Steinhaus, 16 Riesz–Thorin convexity, 9 basic separation, 29, 132 Rockafellar’s, 118, 278 bicontinuity of Attouch–Wets convergence, 290 sandwich, 7, 64, 133 bicontinuity of Mosco convergence, 288 separation, 64, 67, 132, 133 bipolar, 209 Shapley–Folkman, 5, 72 Birkhoff, 8 Sion minimax, 12 Bishop–Phelps, 163, 169 smooth variational principle, 165 Blaschke–Santalo, 32 Šmulian’s, 153–156, 159, 160, 210, 211 Bohr–Mollerup, 9, 32 Stegall’s variational principle, 324 Brodkskii–Mil’man, 167 Straszewicz, 92 Brøndsted–Rockafellar, 121, 162 subdifferential sum rule, 7, 48, 135 Carathéodory, 5, 71 supporting hyperplane, 65 compact separation, 136 theorems of the alternative, 70 Davis, 101 Toeplitz–Hausdorff, 15 Davis–Lewis convexity, 11 von Neumann’s minimax, 12, 74, 186 ∗ Dunford–Pettis, 302 weak -separation, 136 Ekeland’s variational principle, 161, 168 three-slope inequality, 19 in Euclidean space, 117 Tikhonov well-posed, 235

520 Index

topology norm, 228, 274 ∗ bounded weak , 447 characterization, 228 weak, see weak topology duality, 229 ∗ ∗ weak , see weak -topology uniformly integrable set, 301 total, 140 uniformly smooth totally convex function, 374 function, 155, 227, 252, 255 modulus of, 374 and uniformly smooth norms, 247 trace, of an operator, 312 coincidence with uniform Fréchet trace class operators, 312 differentiability, 155 trace norm, 107 duality, 245 transversality conditions, 406 on bounded sets, 264 triangle inequality, 22 versus uniformly smooth on bounded sets, twice Fréchet differentiable, see second-order 249 derivative function on bounded sets, 156 twice Gâteaux differentiable, see second-order duality, 243 derivative modulus, see modulus of smoothness type (ANA), (BR), (D), (ED), (FP), (NI), (VFP), norm, 216, 227 (WD), see maximal montone operator, type and reflexivity, 217 (ANA), type (BR), etc. and superreflexivity, 218, 225 coincidence with uniform Fréchet differentiability, 216 uniform convergence, 31, 115 duality, 216, 217 and conjugation, 297 space, 216 and infimal convolution, 184 uniqueness property, 454 on bounded sets unit ball, 22 and conjugation, 52, 176, 285, 298 and infimal convolution, 52, 176 unit sphere, 22 convergence of subdifferentials, 43 unitary versus Attouch-Wets convergence, 297 equivalent, 313 Yoshida approximation, 52, 176 invariant, 313 uniformly convex upper-semicontinuous function, 127 function, 241, 256, 258 Urysohn lemma, 116 and renorming, 264 USC multifunction, see multifunction, USC at a point, 272, 374 usco, 111, 279 duality, 245 minimal, 279, 283 growth, 244 utility growth and renorming, 260 direct, 56 minimization, 246 maximization, 56 on bounded sets, 264 onto subdifferential, 246 value function, polyhedral, 97 stable under sums, 241 variational principle, see Borwein’s, function on bounded sets, 241, 241 Borwein–Preiss, Ekeland’s, smooth, Stegall’s and uniformly convex norms, 246 (VFP) operator, see maximal monotone operator, duality, 242, 243 type (VFP) modulus, see modulus of convexity von Neumann’s minimax theorem, 12, 74, norm, 215, 216, 221, 226, 246, 254, 186 256, 264 and reflexivity, 217 and superreflexivity, 218, 225 (WD) operator, see maximal monotone operator, duality, 216, 217 type (WD) extensions of, 229 weak Asplund space, 198 preserved by sums, 225 weak Hadamard bornology, 149 space, 168, 215 weak Hadamard differentiability, 150 ∗ ∗ weak , see weak -uniformly convex versus Fréchet differentiability, 379, 381, 384, weakly, see weakly uniformly convex 385 uniformly Fréchet differentiable versus Gâteaux differentiability, 383, 385 coincidence with uniform smoothness, 155 weak topology, 129, 461 function, 155 convergence, 130 ∗ norm, 216 weak -Asplund space, 321 smooth, see uniformly smooth characterization, 325 ∗ uniformly Gâteaux differentiable weak -lower-semicontinuous, 130, 138 bump function, 274 norm, 210 ∗ function, 156 weak -separation theorem, 136 ∗ characterization, 156 weak -topology, 129, 461 duality, 250 convergence, 130

Index 521

∗ weak -uniformly convex Weyl, 472 function, 250 Wijsman convergence, 290 duality, 250 and minimizing sequences, 293 norm, 229 failure to preserve minima, 294 duality, 229 and renorming, 290, 298 weakly compactly generated space, 219 and the Kadec property, 291, 298, 299 Asplund and renorming, 220 not preserved under conjugation, 296 renorms of, 220 versus Kuratowski–Painlevé weakly lower-semicontinuous, 130, 138 convergence, 295 weakly uniformly convex versus Mosco convergence, 296, 299 function, 250 versus slice convergence, 291, 298 duality, 250 norm, 229 zone consistency, 364 duality, 229 Legendre functions, 362, 368