Approximation and Optimization: Algorithms, Complexity and Applications

Preface......Page 5
Contents......Page 7
Contributors......Page 9
 1 Survey......Page 11
 1 Introduction......Page 15
 2 Convex Constraints......Page 17
 3 The General Constrained Case......Page 22
 4 Discussion......Page 33
 References......Page 36
 1 Introduction......Page 37
 2 Example......Page 38
 3 Theoretical Notes......Page 41
 References......Page 43
 1 Introduction......Page 45
 2 Basic Concepts of Multi-Objective Optimization......Page 47
 3 Data Preprocessing......Page 49
 4 Supervised Learning......Page 52
 5 Unsupervised Learning......Page 56
 7 Synopsis and Discussion......Page 58
 References......Page 59
 1 Introduction......Page 66
 2 Early Developments......Page 68
 3 No Free Lunch for Optimization and Search......Page 70
 4 More Recent Work of Wolpert......Page 73
  5.1 No Free Lunches and Evolutionary Algorithms......Page 74
  5.2 No Free Lunches and Meta-Heuristic Techniques......Page 77
 6 NFL for Supervised Learning......Page 83
  6.2 No Free Lunch for Cross-Validation......Page 85
  6.3 Real-World Machine Learning Classification and No Free Lunch Theorems: An Experimental Approach......Page 87
 References......Page 88
 1 Introduction......Page 92
 2 The Algorithm......Page 93
  2.1 The Case q=0......Page 96
  2.2 The Case q=1......Page 101
  2.3 The Case q=2......Page 109
  2.4 The General Case......Page 116
  3.1 Synthetic Test Data......Page 122
  3.2 Real Test Data......Page 125
 References......Page 127
 1 Introduction......Page 128
 2 The Theorem......Page 130
 3 Estimation of Peaks of an NMR Spectrum......Page 139
 References......Page 142
 1 History and Background......Page 144
  2.1 Four Settings......Page 145
  2.2 Study of Optimization Problem......Page 146
 3 Zolotarëv Fraction......Page 147
 4 Projective View......Page 148
  4.1 Projective Problem Setting......Page 149
  4.2 Decomposition into Subclasses......Page 150
 5 Problem Genesis: Signal Processing......Page 151
  6.1 Remez-Type Methods......Page 153
 7 Novel Analytical Approach......Page 154
 8 Examples of Filter Design......Page 155
 References......Page 157
 1 Introduction......Page 159
   2.1.1 Markowitz Model and Its Variations......Page 160
   2.1.2 Single-Factor Model......Page 163
   2.2.1 Estimation of Means......Page 164
   2.2.2 Estimation of Covariances......Page 165
   2.2.3 Ledoit and Wolf Shrinkage Estimator for Covariance Matrix......Page 167
 3 Properties of Selected Portfolios......Page 169
   3.1.1 Real Data......Page 170
   3.1.2 Generated Data......Page 172
  3.2 Bias of Portfolio Returns......Page 174
  3.3 Shrinkage Estimators for Mean Vectors......Page 182
   3.3.1 Improvements from Shrinkage Estimators for Means (a1)......Page 184
   3.3.2 Improvements from Shrinkage Estimators for Means(a2)......Page 185
  3.4 Student t Distribution......Page 189
 4 Future Research......Page 190
 References......Page 191
 1 Introduction......Page 193
 2 Numerical Solution of Multiphysics Problems......Page 195
 3 Structural Mathematical Model for Composites......Page 196
  3.1 Displacement and Strains of the Non-adhesive Layers......Page 197
  3.2 Constitutive Equations of Piezoelectric Layer......Page 198
  3.4 The Adhesive Layer......Page 199
  3.5 Finite Element Formulation......Page 201
  3.6 Variational Principle......Page 202
  3.7 Equations of Motion......Page 206
 4 Modal and Dynamic Analysis......Page 207
  5.1 Fuzzy Control in General......Page 209
  5.2 Development of a Simple Fuzzy Controller......Page 211
  6.1 An Example of a Structural Model with Two Materials......Page 214
  6.2 Fuzzy Control of Smart Plates in the Presence of Delamination......Page 216
   6.2.1 Fuzzy Control on the Non-delaminated Coupled Electromechanical Model......Page 217
   6.2.2 Optimization of Fuzzy Control with Genetic Algorithms......Page 219
   6.2.3 Fuzzy Control on the Delaminated, Coupled Electromechanical Model......Page 221
 7 Conclusions......Page 222
 References......Page 224
 1 Introduction......Page 226
 2 A POMDP Model of Tax Evasion......Page 228
  2.1 The Firm\'s State and Action Sets......Page 229
  2.2 State Evolution......Page 230
   2.2.1 Transition Probabilities......Page 231
  2.4 Firm Observations, Belief, and Value Function......Page 233
  2.5 Solving for the Firm\'S Optimal Policy......Page 235
  3.1 Model Validation: The Case of ``Perfect\'\' Observations......Page 236
  3.2 The Role of Uncertain Observations......Page 237
  3.3 The Role of Statute of Limitations......Page 238
 4 Conclusions......Page 240
  Transition Matrix when Firm Applies for Closure......Page 241
  Transition Matrix when Firm Declines Closure......Page 242
 References......Page 243