Convex analysis and minimization algorithms: 2 Advanced theory and bundle methods
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin u.a.
Springer
1993
|
| Series: | Grundlehren der mathematischen Wissenschaften
306 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Hier auch später erschienene, unveränderte Nachdrucke |
| Physical Description: | XVII, 346 S. Diagramme |
| ISBN: | 0387568522 3540568522 9783642081620 |
Staff View
MARC
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| 245 | 1 | 0 | |a Convex analysis and minimization algorithms |n 2 |p Advanced theory and bundle methods |c Jean-Baptiste Hiriart-Urruty ; Claude Lemaréchal |
| 264 | 1 | |a Berlin u.a. |b Springer |c 1993 | |
| 300 | |a XVII, 346 S. |b Diagramme | ||
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| adam_text | Table of Contents Part II Introduction.......................................... XV IX. Inner Construction of the Subdifferential................................................. 1 1 The Elementary Mechanism.................................................................... 2 2 Convergence Properties........................................................................... 2.1 Convergence..................................................................................... 2.2 Speed of Convergence .......................... 3 Putting the Mechanism in Perspective.................................................... 3.1 Bundling as a Substitute for Steepest Descent................................. 3.2 Bundling as an Emergency Device for Descent Methods ..... 3.3 Bundling as a Separation Algorithm .............................................. 9 9 15 24 24 27 29 X. Conjugacy in Convex Analysis ................................................................. 35 1 The Convex Conjugate of a Function.................................................... 1.1 Definition and First Examples................ .......................................... 1.2 Interpretations................................................................. 1.3 First Properties.................................................................................. 1.4 Subdifferentials of Extended-Valued Functions............. ................ 1.5 Convexification and Subdifferentiability ....................................... 37 37 40 42 47 49 2 Calculus Rules on the Conjugacy
Operation........................................... 2.1 Image of a Function Under a Linear Mapping................................. 2.2 Pre-Composition with an Affine Mapping....................................... 54 54 56 2.3 Sum of Two Functions ........................................................................ 61 2.4 Infima and Suprema........................................................................... 2.5 Post-Composition with an Increasing Convex Function................ 2.6 A Glimpse of Biconjugate Calculus................................................. 3 Various Examples..................................................................................... 3.1 The Cramer Transformation.............................................................. 3.2 Some Results on the Euclidean Distance to a Closed Set............. 3.3 The Conjugate of Convex Partially Quadratic Functions............. 3.4 Polyhedral Functions........................................................................ 4 Differentiability of a Conjugate Function.............................................. 4.1 First-Order Differentiability.............................................................. 4.2 Towards Second-Order Differentiability........................................... 65 69 71 72 72 73 75 76 79 79 82
VI Table of Contents Part II XI. Approximate Subdifferentials of Convex Functions................................. 91 1 The Approximate Subdifferential........................................................... 1.1 Definition, First Properties and Examples........................................ 1.2 Characterization via the Conjugate Function................................. 1.3 Some Useful Properties..................................................................... 92 92 95 98 2 The Approximate Directional Derivative................................................. 102 2.1 The Support Function of the Approximate Subdifferential .... 102 2.2 Properties of the Approximate Difference Quotient.......................... 106 2.3 Behaviour of // and Te as Functions of e........................................... 110 3 Calculus Rules on the Approximate Subdifferential................................. 113 3.1 Sum of Functions.................................................................................. 113 3.2 Pre-Composition with an Affine Mapping........................................... 116 3.3 Image and Marginal Functions........................................................... 118 3.4 A Study of the Intimai Convolution.....................................................119 3.5 Maximum of Functions........................................................................ 123 3.6 Post-Composition with an Increasing Convex Function.................125 4 The Approximate Subdifferential as a Multifunction.............................. 127 4.1 Continuity Properties of the
Approximate Subdifferential............. 127 4.2 Transportation of Approximate Subgradients.................................... 129 XII. Abstract Duality for Practitioners................................................................. 137 1 The Problem and the General Approach.....................................................137 1.1 The Rules of the Game........................................................................ 137 1.2 Examples............................................................................................... 141 2 The Necessary Theory ............................................................................... 147 2.1 Preliminary Results: The Dual Problem.............................................. 147 2.2 First Properties of the Dual Problem................................................. 150 2.3 Primal-Dual Optimality Characterizations ........................................ 154 2.4 Existence of Dual Solutions................................................................. 157 3 Illustrations.................................................................................................161 3.1 The Minimax Point of View..................................................................161 3.2 Inequality Constraints........................................................................... 162 3.3 Dualization of Linear Programs........................................................... 165 3.4 Dualization of Quadratic Programs.....................................................166 3.5 Steepest-Descent Directions
.............................................................. 168 4 Classical Dual Algorithms.......................................................................... 170 4.1 Subgradient Optimization..................................................................... 171 4.2 The Cutting-Plane Algorithm.............................................................. 174 5 Putting the Method in Perspective............................... 178 5.1 The Primal Function........................................................................... 178 5.2 Augmented Lagrangians..................................................................... 181 5.3 The Dualization Scheme in Various Situations................................. 185 5.4 Fenchel’s Duality.................................................................................. 190
Table of Contents Part П VII XIII. Methods of e-Descent..................................................................................... 195 1 Introduction. Identifying the Approximate Subdifferential .................... 195 1.1 The Problem and Its Solution.............................................................. 195 1.2 The Line-Search Function.................................................................... 199 1.3 The Schematic Algorithm.................................................................... 203 2 A Direct Implementation: Algorithm of e-Descent................................ 206 2.1 Iterating the Line-Search.................................................................... 206 2.2 Stopping the Line-Search . . ....................................................... . 209 2.3 The e-Descent Algorithm and Its Convergence................................ 212 3 Putting the Algorithm in Perspective....................................................... 216 3.1 A Pure Separation Form....................................................................... 216 3.2 A Totally Static Minimization Algorithm.......................................... 219 XIV. Dynamic Construction of Approximate Subdifferentials: Dual Form of Bundle Methods ...........................................................223 1 Introduction: The Bundle of Information................................................. 223 1.1 Motivation........................................................................................... 223 1.2 Constructing the Bundle of
Information............................................. 227 2 Computing the Direction.......................................................................... 233 2.1 The Quadratic Program....................................................................... 233 2.2 Minimality Conditions....................................................................... 236 2.3 Directional Derivatives Estimates....................................................... 241 2.4 The Role of the Cutting-Plane Function............................................. 244 3 The Implementable Algorithm ................................................................. 248 3.1 Derivation of the Line-Search..............................................................248 3.2 The Implementable Line-Search and Its Convergence . ...... 250 3.3 Derivation of the Descent Algorithm.................................................254 3.4 The Implementable Algorithm and Its Convergence .......................257 4 Numerical Illustrations.............................................................................. 263 4.1 Typical Behaviour................................................................................. 263 4.2 The Role of £........................................................................................266 4.3 A Variant with Infinite £ : Conjugate Subgradients.......................... 268 4.4 The Role of the Stopping Criterion.................................................... 269 4.5 The Role of Other Parameters..............................................................271 4.6
General Conclusions...........................................................................273 XV. Acceleration of the Cutting-Plane Algorithm: Primal Forms of Bundle Methods........................................................ 275 1 Accelerating the Cutting-Plane Algorithm................................................. 275 1.1 Instability of Cutting Planes................................................................. 276 1.2 Stabilizing Devices: Leading Principles............................................. 279 1.3 A Digression: Step-Control Strategies ............................................. 283 2 A Variety of Stabilized Algorithms.......................................................... 285 2.1 The Trust-Region Point of View ....................................................... 286
vin Table of Contents Part II 2.2 The Penalization Point of View...........................................................289 2.3 The Relaxation Point of View.............................................................. 292 2.4 A Possible Dual Point of View...........................................................295 2.5 Conclusion........................................................................................... 299 3 A Class of Primal Bundle Algorithms....................................................... 301 3.1 The General Method........................................................................... 301 3.2 Convergence........................................................................................ 307 3.3 Appropriate Stepsize Values .............................................................. 314 4 Bundle Methods as Regularizations...........................................................317 4.1 Basic Properties of the Moreau-Yosida Regularization................. 317 4.2 Minimizing the Moreau-Yosida Regularization................................. 322 4.3 Computing the Moreau-Yosida Regularization................................. 326 Bibliographical Comments References Index ..................................................................................... 331 ............................................................................................................... 337 .........................................................................................................................345
Table of Contents Part I Introduction...............................................................................................................XV I. Convex Functions of One Real Variable.................................................... 1 1 Basic Definitions and Examples.............................................................. 1.1 First Definitions of a Convex Function........................................... 1.2 Inequalities with More Than Two Points....................................... 1.3 Modern Definition of Convexity.................................................... 2 First Properties.......................... 2.1 Stability Under Functional Operations........................................... 2.2 Limits of Convex Functions.............................................................. 2.3 Behaviour at Infinity........................................................................ 1 2 6 8 9 9 11 14 3 Continuity Properties.............................................................................. 3.1 Continuity on the Interior of the Domain....................................... 3.2 Lower Semi-Continuity: Closed Convex Functions....................... 3.3 Properties of Closed Convex Functions........................................... 4 First-Order Differentiation .................................................................... 4.1 One-Sided Differentiability of Convex Functions.......................... 4.2 Basic Properties of Subderivatives................................................. 4.3 Calculus
Rules.................................................................................. 5 Second-Order Differentiation................................................................. 5.1 The Second Derivative of a Convex Function................................. 5.2 One-Sided Second Derivatives........................................................ 5.3 How to Recognize a Convex Function.............................................. 6 First Steps into the Theory of Conjugate Functions.............................. 6.1 Basic Properties of the Conjugate.................................................... 6.2 Differentiation of the Conjugate........................................................ 6.3 Calculus Rules with Conjugacy........................................................ 16 16 17 19 20 21 24 27 29 30 32 33 36 38 40 43 II. Introduction to Optimization Algorithms ................................................. 47 1 Generalities............................................................................................. 1.1 The Problem..................................................................................... 1.2 General Structure of Optimization Schemes ................................. 1.3 General Structure of Optimization Algorithms.............................. 2 Defining the Direction.............................................................................. 47 47 50 52 54
X Table of Contents Part I 2.1 Descent and Steepest-Descent Directions....................................... 2.2 First-Order Methods ....................................................................... 2.3 Newtonian Methods.......................................................................... 2.4 Conjugate-Gradient Methods.......................................................... 3 Line-Searches........................................................................................... 3.1 General Structure of a Line-Search................................................. 3.2 Designing the Test (0), (R), (L)....................................................... 3.3 The Wolfe Line-Search.................................................................... 3.4 Updating the Trial Stepsize............................................................. III. Convex Sets................................................................................................. 1 Generalities.............................................................................................. 1.1 Definition and First Examples.......................................................... 1.2 Convexity-Preserving Operations on Sets....................................... 1.3 Convex Combinations and Convex Hulls....................................... 1.4 Closed Convex Sets and Hulls ....................................................... 2 Convex Sets Attached to a Convex Set................................................. 2.1 The Relative
Interior....................................................................... 2.2 The Asymptotic Cone....................................................................... 2.3 Extreme Points................................................................................. 2.4 Exposed Faces ................................................................................. 3 Projection onto Closed Convex Sets....................................................... 3.1 The Projection Operator.................................................................... 3.2 Projection onto a Closed Convex Cone.......................................... 4 Separation and Applications.................................................................... 4.1 Separation Between Convex Sets.................................................... 4.2 First Consequences of the Separation Properties.......................... 4.3 The Lemma of Minkowski-Farkas ................................................. 5 Conical Approximations of Convex Sets .............................................. 5.1 Convenient Definitions of Tangent Cones....................................... 5.2 The Tangent and Normal Cones to a Convex Set.......................... 5.3 Some Properties of Tangent and Normal Cones............................. IV. Convex Functions of Several Variables .................................................... 1 Basic Definitions and Examples.............................................................. 1.1 The Definitions of a Convex Function.............................................. 1.2
Special Convex Functions: Affinity and Closedness....................... 1.3 First Examples.................................................................................. 2 Functional Operations Preserving Convexity....................................... 2.1 Operations Preserving Closedness ................................................. 2.2 Dilations and Perspectives of a Function....................................... 2.3 Infimal Convolution........................................................................... 2.4 Image of a Function Under a Linear Mapping................................. 2.5 Convex Hull and Closed Convex Hull of a Function....................
Table of Contents Part I XI 3 Local and Global Behaviour of a Convex Function................................. 173 3.1 Continuity Properties........................................................................... 173 3.2 Behaviour at Infinity........................................................................... 178 4 First-and Second-Order Differentiation.................................................... 183 4.1 Differentiable Convex Functions........................................................183 4.2 Nondifferentiable Convex Functions................................................. 188 4.3 Second-Order Differentiation.............................................................. 190 V. Sublinearity and Support Functions.................................................... ... . 195 1 Sublinear Functions.................................................................................... 197 1.1 Definitions and First Properties...........................................................197 1.2 Some Examples.................................................................................... 201 1.3 The Convex Cone of All Closed Sublinear Functions....................... 206 2 The Support Function of a Nonempty Set.................................................208 2.1 Definitions, Interpretations................................................................. 208 2.2 Basic Properties.............................................................................. . 211 2.3
Examples.............................................................................................. 215 3 The Isomorphism Between Closed Convex Sets and Closed Sublinear Functions............................................................. 218 3.1 The Fundamental Correspondence.................................................... 218 3.2 Example: Norms and Their Duals, Polarity....................................... 220 3.3 Calculus with Support Functions....................................................... 225 3.4 Example: Support Functions of Closed Convex Polyhedra .... 234 VI. Subdifferentials of Finite Convex Functions.................................................237 1 The Subdifferential: Definitions and Interpretations................................ 238 1.1 First Definition: Directional Derivatives............................................. 238 1.2 Second Definition: Minorization by Affine Functions.......................241 1.3 Geometric Constructions and Interpretations....................................243 1.4 A Constructive Approach to the Existence of a Subgradient . . . 247 2 Local Properties of the Subdifferential.................................................... 249 2.1 First-Order Developments.................................................................... 249 2.2 Minimality Conditions....................................................................... 253 2.3 Mean-Value Theorems....................................................................... 256 3 First Examples
........................................................................................... 258 4 Calculus Rules with Subdifferentials....................................................... 261 4.1 Positive Combinations of Functions.................................................... 261 4.2 Pre-Composition with an Affine Mapping.......................................... 263 4.3 Post-Composition with an Increasing Convex Function of Several Variables....................................................................... 264 4.4 Supremum of Convex Functions....................................................... 266 4.5 Image of a Function Under a Linear Mapping....................................272 5 Further Examples........................................................................................275 5.1 Largest Eigenvalue of a Symmetric Matrix....................................... 275
хп Table of Contents Part I 5.2 Nested Optimization........................................................................... 277 5.3 Best Approximation of a Continuous Function on a Compact Interval.................................................................... 278 6 The Subdifferential as a Multifunction.................................................... 279 6.1 Monotonicity Properties of the Subdififerential................................. 280 6.2 Continuity Properties of the Subdifferential....................................... 282 6.3 Subdifferentials and Limits of Gradients...........................................284 VII. Constrained Convex Minimization Problems: Minimality Conditions, Elements of Duality Theory ....................... 291 1 Abstract Minimality Conditions................................................................. 292 1.1 A Geometric Characterization.............................................................. 293 1.2 Conceptual Exact Penalty.................................................................... 298 2 Minimality Conditions Involving Constraints Explicitly.......................... 301 2.1 Expressing the Normal and Tangent Cones in Terms of the Constraint-Functions.......................................................... 303 2.2 Constraint Qualification Conditions.................................................... 307 2.3 The Strong Slater Assumption...........................................................311 2.4 Tackling the Minimization Problem with Its Data Directly .... 314 3 Properties and Interpretations of the
Multipliers....................................... 317 3.1 Multipliers as a Means to Eliminate Constraints: the Lagrange Function................................................................. 317 3.2 Multipliers and Exact Penalty.............................................................. 320 3.3 Multipliers as Sensitivity Parameters with Respect to Perturbations.............................................................................. 323 4 Minimality Conditions and Saddle-Points................................................. 327 4.1 Saddle-Points: Definitions and First Properties................................. 327 4.2 Mini-Maximization Problems.............................................................. 330 4.3 An Existence Result.............................................................................. 333 4.4 Saddle-Points of Lagrange Functions................................................. 336 4.5 A First Step into Duality Theory....................................................... 338 VIII. Descent Theory for Convex Minimization: The Case of Complete Information....................................................343 1 Descent Directions and Steepest-Descent Schemes................................. 343 1.1 Basic Definitions................................................................................. 343 1.2 Solving the Direction-Finding Problem.............................................. 347 1.3 Some Particular Cases ........................................................................ 351 1.4
Conclusion........................................................................................... 355 2 Illustration. The Finite Minimax Problem................................................. 356 2.1 The Steepest-Descent Method for Finite Minimax Problems . . . 357 2.2 Non-Convergence of the Steepest-Descent Method.......................... 363 2.3 Connection with Nonlinear Programming........................................... 366
Table of Contents Part I XIII 3 The Practical Value of Descent Schemes................................................. 371 3.1 Large Minimax Problems.................................................................... 371 3.2 Infinite Minimax Problems................................................................. 373 3.3 Smooth but Stiff Functions................................................................. 374 3.4 The Steepest-Descent Trajectory....................................................... 377 3.5 Conclusion........................................................................................... 383 Appendix: Notations 1 2 3 4 5 6 .............................................................................................. 385 Some Facts About Optimization................................................................. 385 The Set of Extended Real Numbers.......................................................... 388 Linear and Bilinear Algebra....................................................................... 390 Differentiation in a Euclidean Space ....................................................... 393 Set-Valued Analysis.................................................................................... 396 A Bird’s Eye View of Measure Theory and Integration.......................... 399 Bibliographical Comments References Index .................................................................................... 401 .............................................................................................................. 407
........................................................................................................................ 415
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| author | Hiriart-Urruty, Jean-Baptiste 1949- Lemaréchal, Claude 1944- |
| author_GND | (DE-588)128857102 (DE-588)128857137 |
| author_facet | Hiriart-Urruty, Jean-Baptiste 1949- Lemaréchal, Claude 1944- |
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| classification_rvk | SK 750 |
| classification_tum | MAT 520f MAT 916f MAT 469f |
| ctrlnum | (OCoLC)165173814 (DE-599)BVBBV008552202 |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV008552202 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T17:21:01Z |
| institution | BVB |
| isbn | 0387568522 3540568522 9783642081620 |
| language | English |
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| oclc_num | 165173814 |
| open_access_boolean | |
| owner | DE-12 DE-20 DE-384 DE-91G DE-BY-TUM DE-824 DE-29T DE-739 DE-703 DE-19 DE-BY-UBM DE-706 DE-634 DE-83 DE-11 DE-188 |
| owner_facet | DE-12 DE-20 DE-384 DE-91G DE-BY-TUM DE-824 DE-29T DE-739 DE-703 DE-19 DE-BY-UBM DE-706 DE-634 DE-83 DE-11 DE-188 |
| physical | XVII, 346 S. Diagramme |
| publishDate | 1993 |
| publishDateSearch | 1993 |
| publishDateSort | 1993 |
| publisher | Springer |
| record_format | marc |
| series | Grundlehren der mathematischen Wissenschaften |
| series2 | Grundlehren der mathematischen Wissenschaften |
| spelling | Hiriart-Urruty, Jean-Baptiste 1949- Verfasser (DE-588)128857102 aut Convex analysis and minimization algorithms 2 Advanced theory and bundle methods Jean-Baptiste Hiriart-Urruty ; Claude Lemaréchal Berlin u.a. Springer 1993 XVII, 346 S. Diagramme txt rdacontent n rdamedia nc rdacarrier Grundlehren der mathematischen Wissenschaften 306 Grundlehren der mathematischen Wissenschaften ... Hier auch später erschienene, unveränderte Nachdrucke Konvexe Analysis (DE-588)4138566-4 gnd rswk-swf Konvexe Analysis (DE-588)4138566-4 s DE-604 Lemaréchal, Claude 1944- Verfasser (DE-588)128857137 aut (DE-604)BV008398102 2 Erscheint auch als Online-Ausgabe 978-3-662-06409-2 Grundlehren der mathematischen Wissenschaften 306 (DE-604)BV000000395 306 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005621410&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hiriart-Urruty, Jean-Baptiste 1949- Lemaréchal, Claude 1944- Convex analysis and minimization algorithms Grundlehren der mathematischen Wissenschaften Konvexe Analysis (DE-588)4138566-4 gnd |
| subject_GND | (DE-588)4138566-4 |
| title | Convex analysis and minimization algorithms |
| title_auth | Convex analysis and minimization algorithms |
| title_exact_search | Convex analysis and minimization algorithms |
| title_full | Convex analysis and minimization algorithms 2 Advanced theory and bundle methods Jean-Baptiste Hiriart-Urruty ; Claude Lemaréchal |
| title_fullStr | Convex analysis and minimization algorithms 2 Advanced theory and bundle methods Jean-Baptiste Hiriart-Urruty ; Claude Lemaréchal |
| title_full_unstemmed | Convex analysis and minimization algorithms 2 Advanced theory and bundle methods Jean-Baptiste Hiriart-Urruty ; Claude Lemaréchal |
| title_short | Convex analysis and minimization algorithms |
| title_sort | convex analysis and minimization algorithms advanced theory and bundle methods |
| topic | Konvexe Analysis (DE-588)4138566-4 gnd |
| topic_facet | Konvexe Analysis |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005621410&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV008398102 (DE-604)BV000000395 |
| work_keys_str_mv | AT hiriarturrutyjeanbaptiste convexanalysisandminimizationalgorithms2 AT lemarechalclaude convexanalysisandminimizationalgorithms2 |