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xix | |
| Preface |
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xxi | |
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Chapter 1 Semilimits and semicontinuity of multimappings |
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1 | (64) |
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1.1 Generalities on multimappings |
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2 | (2) |
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1.2 Semilimits of multimappings |
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4 | (12) |
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1.3 Epigraphical semilimits |
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16 | (3) |
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1.4 Semicontinuity of multimappings |
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19 | (5) |
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1.5 Scalarization of semicontinuity |
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24 | (9) |
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1.6 Semicontinuity of sum and convexification |
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33 | (1) |
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1.7 Michael continuous selection theorem |
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34 | (2) |
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1.8 Hausdorff-Pompeiu semidistances |
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36 | (12) |
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1.8.1 Hausdorff-Pompeiu excess and distance |
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37 | (2) |
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1.8.2 Truncated Hausdorff-Pompeiu excess and distance |
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39 | (6) |
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1.8.3 Hausdorff and Attouch-Wets semicontinuities |
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45 | (2) |
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1.8.4 Holder continuity of metric projection with convex set as variable |
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47 | (1) |
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48 | (8) |
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1.9.1 Further properties of semicontinuous multimappings |
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48 | (2) |
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1.9.2 Hausdorff-Pompeiu distance between boundaries of sets |
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50 | (3) |
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1.9.3 Distances between cones |
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53 | (3) |
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56 | (9) |
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Chapter 2 Tangent cones and Clarke subdifferential |
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65 | (184) |
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2.1 Clarke tangent and normal cones |
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65 | (18) |
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2.1.1 Definitions and various characterizations |
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65 | (7) |
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2.1.2 Interior tangent cone and calculus for Clarke tangent and normal cones of intersection |
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72 | (4) |
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2.1.3 Epi-Lipschitz sets; their geometrical, tangential and topological properties |
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76 | (7) |
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2.2 Clarke subdifferential |
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83 | (73) |
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2.2.1 C-subdifferential: definition and examples |
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83 | (1) |
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2.2.2 Differentiability and strict differentiability |
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84 | (9) |
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2.2.3 C-subdifferential, minimizers and derivatives |
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93 | (2) |
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2.2.4 C-subdifferential and convex functions |
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95 | (8) |
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2.2.5 C-subdifferential and locally Lipschitz functions |
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103 | (10) |
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2.2.6 Gradient representation of C-subdifferential of locally Lipschitz functions |
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113 | (2) |
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2.2.7 Clarke tangent and normal cones of epigraphs and graphs |
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115 | (3) |
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2.2.8 C-subdifferential and directionally Lipschitz functions |
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118 | (3) |
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2.2.9 Clarke tangent and normal cones through distance function |
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121 | (2) |
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2.2.10 Rockafellar theorem for C-subdifferential of finite sum of functions |
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123 | (5) |
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2.2.11 Bouligand-Peano tangent cone and Bouligand directional derivative |
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128 | (6) |
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2.2.12 Tangential regularity of sets and functions |
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134 | (6) |
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2.2.13 Tangent cones of inverse and direct images |
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140 | (8) |
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2.2.14 Tangential regularity of sets versus tangential regularity of distance functions |
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148 | (3) |
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2.2.15 Signed distance function and Clarke tangent cone |
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151 | (3) |
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2.2.16 Sublevel representation of epi-Lipschitz sets |
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154 | (2) |
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2.3 Local Lipschitz property of continuous convex functions |
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156 | (14) |
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2.3.1 Lipschitz property of convex functions |
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156 | (5) |
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2.3.2 Applications to directional Lipschitz property |
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161 | (1) |
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2.3.3 Lipschitz property of continuous vector-valued convex mappings |
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162 | (8) |
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2.4 Chain rule, compactly Lipschitzian mappings, supremum functions |
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170 | (24) |
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2.4.1 Compactly Lipschitzian mappings and chain rule |
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170 | (9) |
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2.4.2 C-subdifferential of supremum of finitely many functions, extension of Lipschitz mappings, tangent cones of sublevel sets |
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179 | (6) |
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2.4.3 Clarke theorem for C-subdifferential of supremum of infinitely many Lipschitz functions |
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185 | (3) |
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2.4.4 Valadier theorem for suprema of infinitely many convex functions in normed spaces |
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188 | (3) |
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2.4.5 C-subgradients of distance function through metric projection; nonzero C-normals at boundary points |
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191 | (3) |
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2.5 Optimization problems with constraints |
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194 | (11) |
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2.5.1 Penalization principles with Lipschitz/non-Lipschitz objective functions |
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194 | (3) |
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2.5.2 Minimization under a set-constraint |
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197 | (1) |
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2.5.3 Ekeland variational principle and Bishop-Phelps principles |
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198 | (4) |
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2.5.4 General optimization problems |
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202 | (3) |
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2.6 Clarke tangent cone in terms of Bouligand-Peano tangent cones |
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205 | (4) |
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2.6.1 Danes' drop theorem |
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205 | (2) |
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2.6.2 C-tangent cone and limit inferior of B-tangent cones |
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207 | (2) |
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2.7 Basic tangential properties through measure theory |
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209 | (19) |
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2.7.1 Points of nullity of symmetrized B-tangent cone |
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209 | (3) |
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212 | (3) |
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2.7.3 Metrics on the set of vector subspaces |
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215 | (4) |
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2.7.4 Points of nullity of trace of B-tangent cone on subspace |
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219 | (1) |
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2.7.5 Tangential properties through Hausdorff measure |
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220 | (8) |
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228 | (8) |
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2.8.1 Intersection of C-normal cones |
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228 | (2) |
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2.8.2 Subset of a Cartesian product |
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230 | (2) |
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2.8.3 Compactly epi-Lipschitz sets |
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232 | (4) |
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236 | (13) |
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Chapter 3 Convexity and duality in locally convex spaces |
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249 | (224) |
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3.1 Convex functions on topological vector spaces |
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249 | (28) |
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3.1.1 Subdifferential and directional derivatives of convex functions on topological vector spaces |
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249 | (7) |
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3.1.2 Topological and Lipschitz properties of convex functions on topological vector spaces |
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256 | (4) |
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3.1.3 Lipschitz property of convex functions under growth conditions |
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260 | (3) |
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3.1.4 Coercive convex functions |
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263 | (1) |
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3.1.5 Subdifferentiability and topological properties of subdifferential of convex functions |
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264 | (4) |
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3.1.6 Subdifferential of suprema of infinitely many convex functions in locally convex spaces |
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268 | (4) |
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3.1.7 Subdifferential properties of one variable convex functions |
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272 | (5) |
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3.2 Subdifferentiability of convex functions in finite dimensions |
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277 | (4) |
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3.2.1 Subdifferentiability via the relative interior |
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277 | (1) |
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3.2.2 Subdifferentiability of polyhedral convex functions |
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278 | (3) |
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3.3 Conjugates in the locally convex setting |
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281 | (22) |
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3.3.1 General properties and examples of Legendre-Fenchel conjugate |
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281 | (12) |
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3.3.2 Pointwise supremum of continuous affine functions |
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293 | (2) |
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3.3.3 Biconjugate and Fenchel-Moreau theorem |
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295 | (6) |
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3.3.4 Dual conditions for coercivity of convex functions |
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301 | (1) |
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3.3.5 Global Lipschitz property of conjugate functions |
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302 | (1) |
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3.4 B-differentiability and continuity of subdifferential |
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303 | (15) |
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3.4.1 B-differentiability: Definition and intrinsic characterizations for convex functions |
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304 | (4) |
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3.4.2 B-differentiability of convex functions and continuous selections of subdifferentials |
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308 | (2) |
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3.4.3 B-differentiability of convex functions and continuity of their subdifferentials |
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310 | (6) |
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3.4.4 Lipschitz continuity of e-subdifferential |
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316 | (2) |
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3.5 Asymptotic functions and cones |
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318 | (9) |
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3.5.1 Definitions and general properties |
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318 | (6) |
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3.5.2 Asymptotic functions under convexity |
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324 | (3) |
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3.6 Brønsted-Rockafellar theorem |
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327 | (3) |
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3.7 Duality for the sum with a linear function |
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330 | (2) |
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3.8 Duality in convex optimization |
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332 | (15) |
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3.9 Duality, infsup property, Lagrange multipliers |
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347 | (2) |
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3.10 Linear optimization problem |
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349 | (2) |
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3.11 Sum and chain rules of subdifferential under convexity |
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351 | (8) |
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3.12 Application to chain rule for C-subgradients and C-normals |
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359 | (1) |
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3.13 Calculus rules for normals and tangents to convex sets |
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360 | (4) |
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3.14 Chain rule with partially nondecreasing functions |
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364 | (2) |
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3.15 Extended rules for subdifferential of maximum of finitely many convex functions |
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366 | (4) |
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3.16 Limiting formulas for subdifferential of convex functions |
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370 | (9) |
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3.16.1 Limiting sum/chain rule for subdifferential of convex functions |
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370 | (7) |
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3.16.2 Limiting rules for subdifferential of composition with inner vector-valued convex mapping |
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377 | (2) |
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3.17 Subdifferential determination and maximal monotonicity for convex functions on Banach spaces |
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379 | (2) |
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3.18 Normals to convex sublevel sets |
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381 | (20) |
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3.18.1 Normals to convex sublevels under Slater condition in locally convex spaces |
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381 | (2) |
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3.18.2 Horizon subgradients of convex functions |
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383 | (2) |
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3.18.3 Limiting formulas for normals to convex sublevels: Reflexive Banach space case |
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385 | (4) |
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3.18.4 Limiting formulas for normals to convex sublevels: General Banach space case |
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389 | (2) |
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3.18.5 Limiting formulas for normals to intersection of finitely many sublevels |
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391 | (8) |
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3.18.6 Limiting formulas for normals to vector convex sublevels |
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399 | (2) |
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3.19 Continuity of conjugate functions, weak compactness of sublevels, minimum attainment |
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401 | (15) |
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3.19.1 Continuity of conjugate functions and weak compactness of sublevels |
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401 | (5) |
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3.19.2 Attainment of the minimum off --- <x*, ·> |
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406 | (10) |
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3.20 Subdifferential of distance functions from convex sets |
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416 | (3) |
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3.21 Moreau envelope, strongly convex functions |
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419 | (16) |
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419 | (11) |
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3.21.2 Strongly convex functions |
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430 | (5) |
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3.22 Gateaux differentiability at subdifferentiability points |
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435 | (1) |
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436 | (17) |
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3.23.1 Duality with partial conjugate |
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436 | (2) |
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3.23.2 Calculus for ε-subdifferential of convex functions |
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438 | (3) |
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3.23.3 Extended calculus for ε-subdifferential of convex functions |
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441 | (7) |
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3.23.4 ε-Subdifferential determination of convex functions and cyclic monotonicity |
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448 | (3) |
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3.23.5 Limiting subdifferential chain rule for convex functions on locally convex spaces |
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451 | (2) |
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453 | (20) |
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Chapter 4 Mordukhovich limiting normal cone and subdifferential |
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473 | (118) |
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4.1 Frechet normal and subgradient |
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473 | (20) |
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4.1.1 Definitions and first properties |
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473 | (15) |
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4.1.2 Frechet subgradients of distance functions |
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488 | (5) |
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4.2 Separable reduction principle for F-subdifferentiability |
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493 | (8) |
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494 | (3) |
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4.2.2 Separable reduction of Frechet subdifferentiability |
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497 | (4) |
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4.3 Fuzzy calculus rules for Frechet subdifferentials |
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501 | (22) |
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4.3.1 Borwein-Preiss variational principle |
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502 | (6) |
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4.3.2 Fuzzy sum rule for Frechet subdifferential under Frechet differentiable renorm |
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508 | (4) |
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4.3.3 Applications to convex functions and Asplund spaces |
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512 | (3) |
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4.3.4 Fuzzy sum rule for Frechet subdifferential in Asplund space |
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515 | (3) |
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4.3.5 Fuzzy chain rule for Frechet subdifferential |
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518 | (2) |
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4.3.6 Stegall variational principle, Frechet derivative of conjugate function |
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520 | (3) |
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4.4 Mordukhovich limiting subdifferential in Asplund space |
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523 | (34) |
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4.4.1 Definitions, properties, calculus |
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523 | (4) |
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527 | (8) |
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4.4.3 L-Subdifferential of distance function |
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535 | (5) |
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4.4.4 Mordukhovich limiting subdifferential in normed space |
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540 | (17) |
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4.5 Representation of C-subdifferential via limiting subgradients |
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557 | (6) |
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4.5.1 Horizon L-subgradient and representation of C-subdifferential |
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557 | (4) |
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4.5.2 Analytic description of horizon limiting subgradient |
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561 | (2) |
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4.6 Proximal normal cone and subdifferential |
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563 | (17) |
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4.6.1 Definition and properties of proximal subgradient |
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563 | (10) |
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4.6.2 Proximal subgradients of distance functions |
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573 | (3) |
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4.6.3 Proximal fuzzy calculus and proximal representation of the limiting subdifferential |
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576 | (4) |
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580 | (3) |
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4.7.1 F-normal cone to graphs of multimappings |
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580 | (2) |
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4.7.2 L-subdifferential versus C-subdifferential in the real line |
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582 | (1) |
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583 | (8) |
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Chapter 5 Ioffe approximate subdifferential |
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591 | (56) |
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591 | (8) |
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591 | (6) |
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5.1.2 Hadamard subdifferential of sums of functions |
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597 | (2) |
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5.2 Ioffe A-subdifferential on separable Banach spaces |
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599 | (8) |
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5.2.1 Definition for Lipschitz functions and comparisons |
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599 | (3) |
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5.2.2 A-normal cone in separable Banach spaces |
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602 | (5) |
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5.3 Ioffe A-subdifferential of Lipschitz functions on Banach spaces |
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607 | (9) |
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5.3.1 Definition, properties and sum |
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607 | (9) |
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5.4 A-normal cone and A-subdifferential of general functions |
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616 | (20) |
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5.4.1 A-normal cone in general Banach spaces |
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616 | (6) |
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5.4.2 A-subdifferential of general functions |
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622 | (5) |
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5.4.3 Chain rule for A-subdifferential and mean value inequality |
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627 | (4) |
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5.4.4 Representation of C-subdifferential with A-subgradients |
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631 | (1) |
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5.4.5 Extended A-subdifferential sum rule |
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632 | (4) |
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636 | (7) |
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5.5.1 A-normals to compactly epi-Lipschitz sets |
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636 | (2) |
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5.5.2 Subdifferentially pathological Lipschitz functions |
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638 | (5) |
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643 | (4) |
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Chapter 6 Sequential mean value inequalities |
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647 | (68) |
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6.1 Mean value inequalities with Dini derivatives |
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647 | (31) |
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6.1.1 Mean value inequalities with lower/upper Dini directional derivatives |
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647 | (3) |
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6.1.2 Sub-sup regularity and saddle functions |
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650 | (5) |
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6.1.3 Extended gradient representations of subdifferentials |
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655 | (9) |
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6.1.4 Conditions for monotonicity and other properties via Dini semiderivates |
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664 | (3) |
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6.1.5 Mean value inequality for images of sets and Denjoy-Young-Saks theorem |
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667 | (7) |
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6.1.6 Mean value inequality with Dini subgradients |
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674 | (4) |
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6.2 Zagrodny mean value inequality |
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678 | (16) |
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6.2.1 Density properties for subdifferentials |
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680 | (1) |
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6.2.2 Zagrodny mean value theorem |
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681 | (3) |
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6.2.3 Subdifferential and tangential characterizations of Lipschitz properties |
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684 | (6) |
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6.2.4 Subdifferential and tangential characterizations of monotonicity and convexity |
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690 | (4) |
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6.3 Approximate and sequential Rolle-type theorems |
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694 | (5) |
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6.4 Multidirectional mean value inequalities |
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699 | (12) |
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711 | (4) |
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Chapter 7 Metric regularity |
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715 | (178) |
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7.1 Aubin-Lipschitz property and metric regularity |
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715 | (11) |
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7.2 Openness and metric regularity of convex multimappings: Robinson-Ursescu theorem |
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726 | (2) |
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7.3 Criteria and estimates of rates of openness and metric regularity of multimappings |
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728 | (9) |
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7.4 Metrically regular transversality of system of sets |
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737 | (13) |
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7.5 Metric regularity of convex feasible sets, Hoffman inequality |
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750 | (6) |
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7.6 Metric regularity and Lipschitz additive perturbation |
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756 | (6) |
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7.7 Optimality conditions and calculus of tangent and normal cones under metric subregularity |
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762 | (16) |
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7.7.1 Optimality conditions under metric subregularity or other conditions |
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762 | (2) |
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7.7.2 Estimates of coderivatives under metric subregularity or other conditions; regularity of nonsmooth constraints |
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764 | (3) |
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7.7.3 General optimality conditions |
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767 | (2) |
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7.7.4 Normal/tangent cone calculus and chain rule |
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769 | (9) |
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7.8 More on subdifferential calculus for convex functions |
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778 | (2) |
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780 | (30) |
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7.9.1 Metric subregularity of polyhedral multimappings |
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780 | (2) |
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7.9.2 Metric regularity/subregularity of subdifferential and growth conditions |
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782 | (28) |
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810 | (83) |
| Appendix A Topology |
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817 | (4) |
| Appendix B Topological properties of convex sets |
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821 | (8) |
| Appendix C Functional analysis |
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829 | (6) |
| Appendix D Measure theory |
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835 | (2) |
| Appendix E Differential calculus and differentiable manifolds |
|
837 | (6) |
| Bibliography |
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843 | (36) |
| Index |
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879 | (634) |
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|
xix | |
| Preface |
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xxi | |
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Chapter 8 Subsmooth functions and sets |
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893 | (96) |
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8.1 Definition and first properties of subsmooth functions |
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893 | (9) |
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8.2 Directional derivatives and subdifferentials of subsmooth functions |
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902 | (20) |
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8.2.1 General properties of derivatives and subdifferentials |
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902 | (7) |
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8.2.2 Submonotonicity of subdifferentials |
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909 | (8) |
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8.2.3 Subdifferential characterizations of one-sided subsmooth functions |
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917 | (5) |
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922 | (11) |
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8.3.1 Definition of subsmooth sets and general properties |
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922 | (8) |
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8.3.2 Subsmoothness of sets versus Shapiro property |
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930 | (3) |
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8.4 Epi-Lipschitz subsmooth sets |
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933 | (6) |
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8.5 Metrically subsmooth sets |
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939 | (11) |
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8.6 Subsmoothness of a set and a-far property of the C-subdifferential of its distance function |
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950 | (5) |
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8.7 Preservation of subsmoothness under operations |
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955 | (12) |
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8.8 Metric subregularity under metric subsmoothness |
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967 | (6) |
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8.9 Equi-subsmoothness of sets and subdifferential of their distance functions |
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973 | (5) |
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978 | (7) |
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8.10.1 ε-Localization of subsmooth functions by convex functions |
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979 | (2) |
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8.10.2 Metric regularity of subsmooth-like multimappings |
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981 | (4) |
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985 | (4) |
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Chapter 9 Subdifferential determination |
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989 | (28) |
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989 | (4) |
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9.2 Subdifferentially and directionally stable functions |
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993 | (10) |
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9.2.1 Subdifferentially and directionally stable functions, properties and examples |
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994 | (5) |
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9.2.2 Subdifferential determination of subdifferentially and directionally stable functions |
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999 | (4) |
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9.3 Essentially directionally smooth functions and their subdifferential determination |
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1003 | (10) |
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9.3.1 Essentially directionally smooth functions, properties and examples |
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1003 | (6) |
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9.3.2 Subdifferential determination of essentially directionally smooth functions |
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1009 | (4) |
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1013 | (4) |
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Chapter 10 Semiconvex functions |
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1017 | (52) |
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10.1 Semiconvex functions |
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1017 | (17) |
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10.1.1 Semiconvexity, moduli of semiconvexity |
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1017 | (3) |
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10.1.2 Semiconvexity of diverse types of functions |
|
|
1020 | (3) |
|
10.1.3 Sup-representation of linearly semiconvex functions |
|
|
1023 | (5) |
|
10.1.4 Composite stability for semiconvexity and distance function |
|
|
1028 | (4) |
|
10.1.5 Lipschitz continuity of semiconvex functions |
|
|
1032 | (2) |
|
10.2 Subdifferentials and derivatives of semiconvex functions |
|
|
1034 | (13) |
|
10.2.1 Directional derivatives and subdifferentials |
|
|
1034 | (8) |
|
10.2.2 Properties under linear semiconvexity and linear semiconcavity |
|
|
1042 | (3) |
|
10.2.3 Subdifferential and tangential characterizations of semiconvex functions |
|
|
1045 | (2) |
|
10.3 Max-representation and extension of Lipschitz semiconvex functions |
|
|
1047 | (16) |
|
10.3.1 Max-representation with quadratic/differentiable functions |
|
|
1048 | (2) |
|
10.3.2 Max-representation in uniformly convex space |
|
|
1050 | (13) |
|
10.4 Semiconvex multimappings |
|
|
1063 | (2) |
|
|
|
1065 | (4) |
|
Chapter 11 Primal lower regular functions and prox-regular functions |
|
|
1069 | (62) |
|
11.1 S-Lower regular functions |
|
|
1069 | (17) |
|
11.1.1 Primal lower and s-lower regular functions |
|
|
1069 | (3) |
|
11.1.2 Convexly composite functions |
|
|
1072 | (6) |
|
11.1.3 Coincidence of subdifferentials of s-lower regular functions |
|
|
1078 | (3) |
|
11.1.4 Subdifferential characterization of s-lower regular functions |
|
|
1081 | (5) |
|
|
|
1086 | (13) |
|
11.3 Moreau envelope of primal lower regular functions in Hilbert spaces |
|
|
1099 | (14) |
|
11.3.1 First properties related to continuity of proximal mapping |
|
|
1100 | (1) |
|
11.3.2 Differentiability properties of Moreau envelope of primal lower regular functions |
|
|
1101 | (12) |
|
11.4 Subdifferential determination of primal lower regular functions |
|
|
1113 | (3) |
|
11.5 Prox-regular functions |
|
|
1116 | (9) |
|
11.5.1 Definition and examples |
|
|
1116 | (1) |
|
11.5.2 Subdifferential characterization of prox-regular functions |
|
|
1117 | (5) |
|
11.5.3 Differentiability of Moreau envelopes under prox-regularity |
|
|
1122 | (3) |
|
|
|
1125 | (6) |
|
Chapter 12 Singular points of nonsmooth functions |
|
|
1131 | (22) |
|
12.1 Singular points of nonsmooth mappings |
|
|
1131 | (9) |
|
12.2 Singular points of convex and semiconvex functions |
|
|
1140 | (10) |
|
|
|
1150 | (3) |
|
Chapter 13 Non-differentiability points of functions on separable Banach spaces |
|
|
1153 | (34) |
|
13.1 Non-differentiability points of subregular functions |
|
|
1153 | (1) |
|
13.2 Null sets in infinite dimensions |
|
|
1153 | (16) |
|
13.2.1 Aronszajn null sets |
|
|
1154 | (2) |
|
|
|
1156 | (7) |
|
|
|
1163 | (6) |
|
13.3 Hadamard non-differentiability points of Lipschitz functions |
|
|
1169 | (9) |
|
13.3.1 Hadamard non-differentiability points of Lipschitz mappings |
|
|
1169 | (6) |
|
13.3.2 More on interior tangent property via signed distance function |
|
|
1175 | (1) |
|
13.3.3 Non-differentiability points of one-sided Lipschitz functions |
|
|
1176 | (2) |
|
13.4 Zajicek extension of Denjoy-Young-Saks theorem |
|
|
1178 | (6) |
|
|
|
1184 | (3) |
|
Chapter 14 Distance function, metric projection, Moreau envelope |
|
|
1187 | (40) |
|
14.1 Distance function and metric projection |
|
|
1187 | (21) |
|
14.1.1 Density of points with nearest/farthest points |
|
|
1187 | (6) |
|
14.1.2 Differentiability of distance functions and farthest distance functions under differentiable norms |
|
|
1193 | (10) |
|
14.1.3 Genericity of points with nearest points, Lau theorem |
|
|
1203 | (5) |
|
14.2 Genericity attainment and other properties of Moreau envelopes |
|
|
1208 | (12) |
|
14.3 I Subdifferential by means of Moreau envelopes |
|
|
1220 | (3) |
|
|
|
1223 | (4) |
|
Chapter 15 Prox-regularity of sets in Hilbert spaces |
|
|
1227 | (102) |
|
15.1 P(-)-prox-regularity of sets |
|
|
1227 | (27) |
|
15.2 Uniform and local prox-regularity |
|
|
1254 | (41) |
|
15.2.1 Uniform prox-regularity |
|
|
1254 | (13) |
|
15.2.2 Uniform prox-regularity of r-enlargement and r-exterior set |
|
|
1267 | (6) |
|
15.2.3 Linear semiconvexity of distance function to a prox-regular set |
|
|
1273 | (4) |
|
15.2.4 Uniform prox-regularity of connected components |
|
|
1277 | (1) |
|
15.2.5 Local (r,α)-prox-regularity |
|
|
1278 | (12) |
|
15.2.6 Directional derivability of the metric projection |
|
|
1290 | (5) |
|
|
|
1295 | (1) |
|
15.4 Prox-regularity in operations |
|
|
1296 | (16) |
|
15.4.1 Uniform prox-regularity in operations |
|
|
1297 | (9) |
|
15.4.2 Local prox-regularity in operations |
|
|
1306 | (6) |
|
15.5 Continuity properties of C Pc(u) |
|
|
1312 | (4) |
|
|
|
1316 | (4) |
|
15.6.1 Representation of multimappings with prox-regular values |
|
|
1316 | (2) |
|
15.6.2 Continuous selections of lower semicontinuous multimappings with prox-regular values |
|
|
1318 | (2) |
|
|
|
1320 | (9) |
|
Chapter 16 Compatible parametrization and Vial property of prox-regular sets, exterior sphere condition |
|
|
1329 | (34) |
|
16.1 Compatible parametrization of prox-regular sets |
|
|
1329 | (5) |
|
16.2 Strongly convex sets and Vial property of prox-regular sets |
|
|
1334 | (15) |
|
16.2.1 Strongly convex sets |
|
|
1334 | (7) |
|
|
|
1341 | (5) |
|
16.2.3 Closedness of Minkowski sums and ball separations properties |
|
|
1346 | (3) |
|
16.3 Exterior/interior sphere condition |
|
|
1349 | (10) |
|
|
|
1359 | (4) |
|
Chapter 17 Differentiability of metric projection onto prox-regular sets |
|
|
1363 | (58) |
|
17.1 Further properties of (r,α)-prox-regularity |
|
|
1363 | (13) |
|
17.2 Differentiability of metric projection |
|
|
1376 | (19) |
|
17.2.1 Variational and prox-regularity properties of submanifolds |
|
|
1376 | (8) |
|
17.2.2 Smoothness of metric projection onto prox-regular sets with smooth boundary |
|
|
1384 | (11) |
|
17.3 Characterization of epi-Lipschitz sets with smooth boundary |
|
|
1395 | (11) |
|
17.3.1 Properties of derivatives of metric projection |
|
|
1395 | (4) |
|
17.3.2 Smoothness of the boundary of a set via the metric projection |
|
|
1399 | (7) |
|
17.4 Metric projection onto submanifold |
|
|
1406 | (12) |
|
17.4.1 Differentiability of metric projection onto submanifold |
|
|
1407 | (2) |
|
17.4.2 Characterization of submanifolds via metric projection |
|
|
1409 | (5) |
|
17.4.3 Smoothness property of signed distance function |
|
|
1414 | (4) |
|
|
|
1418 | (3) |
|
Chapter 18 Prox-regularity of sets in uniformly convex Banach spaces |
|
|
1421 | (92) |
|
18.1 Uniformly convex Banach spaces |
|
|
1421 | (42) |
|
18.1.1 Strictly convex normed spaces |
|
|
1421 | (5) |
|
18.1.2 Uniformly convex Banach spaces |
|
|
1426 | (7) |
|
18.1.3 Uniformly smooth Banach spaces |
|
|
1433 | (10) |
|
18.1.4 Characterizations of uniformly convex/smooth norms via duality mappings |
|
|
1443 | (7) |
|
18.1.5 Xu-Roach theorems on moduli of convexity and smoothness |
|
|
1450 | (13) |
|
18.2 Proximal normals in normed spaces |
|
|
1463 | (6) |
|
18.3 Prox-regular sets and J-plr functions |
|
|
1469 | (8) |
|
18.4 Local Moreau envelopes of J-plr functions |
|
|
1477 | (7) |
|
18.4.1 Frechet differentiability of local Moreau envelope |
|
|
1478 | (2) |
|
18.4.2 Uniform continuity of local proximal mappings of J-plr functions |
|
|
1480 | (4) |
|
18.5 Characterizations of local prox-regular sets in uniformly convex Banach spaces |
|
|
1484 | (11) |
|
18.5.1 Metric projection of local prox-regular sets |
|
|
1484 | (5) |
|
18.5.2 Basic characterizations of local prox-regularity in uniformly convex Banach spaces |
|
|
1489 | (4) |
|
18.5.3 Tangential regularity |
|
|
1493 | (2) |
|
18.6 Characterizations and properties of uniformly prox-regular sets in uniformly convex Banach spaces |
|
|
1495 | (9) |
|
18.6.1 Characterizations of uniform prox-regularity of sets in uniformly convex Banach spaces |
|
|
1495 | (6) |
|
18.6.2 Connected components of r-prox-regular sets |
|
|
1501 | (1) |
|
18.6.3 Enlargements and exterior points of r-prox-regular sets |
|
|
1502 | (2) |
|
18.7 Lipschitz continuity of metric projection and radius of prox-regularity |
|
|
1504 | (3) |
|
18.8 Prox-regularity and geometric variational properties of cones |
|
|
1507 | (1) |
|
|
|
1508 | (5) |
| Appendix A Topology |
|
1513 | (4) |
| Appendix B Topological properties of convex sets |
|
1517 | (8) |
| Appendix C Functional analysis |
|
1525 | (6) |
| Appendix D Measure theory |
|
1531 | (2) |
| Appendix E Differential calculus and differentiable manifolds |
|
1533 | (6) |
| Bibliography |
|
1539 | (36) |
| Index |
|
1575 | |
