Optimization, Simulation, and Control

Cover......Page 1
Optimization, Simulation,and Control......Page 4
Semilocal Convergence with R-Order Three Theoremsfor the Chebyshev Method and Its Modifications......Page 8
Appendix: Mathematical Preliminaries......Page 12
1 Introduction......Page 15
2 Motivation and Literature......Page 16
3 The Composition of Convex Envelopes......Page 18
3.1 Alphabets, Languages, and Grammars......Page 19
3.2 Mathematical Expression Language: Syntax......Page 20
3.3.1 Exact Semantics......Page 21
3.3.2 Relaxed Semantics......Page 23
4 Computational Results......Page 25
5 Conclusion......Page 27
References......Page 28
1 Introduction......Page 30
2 Mathematical preliminaries......Page 31
3 Gap functions for vector variational inequalities......Page 34
4.1 Vector variational inequalities with set-valued mappings......Page 43
4.2 Vector variational-like inequalities with set-valued mappings......Page 45
4.3 Generalized vector variational-like inequalities with set-valued mappings......Page 46
References......Page 47
1 Introduction......Page 48
2 Optimal Inscribing of Two Balls......Page 49
3 Continuity and Differentiability of Auxiliary Functions......Page 54
4 Numerical Examples......Page 57
References......Page 59
1 Variational Inequality Problem......Page 62
1.1 Existence and Convexity of the Solution Set of VIP......Page 64
1.3 Traffic Equilibrium......Page 65
2 Mathematical Programs with Equilibrium Constraints......Page 66
3  Methods for Solving the MPEC......Page 67
3.1 Penalty Techniques......Page 68
3.2 Nondifferential Optimization......Page 69
3.3 Smoothing Methods......Page 70
4 Optimality Conditions for MPEC......Page 71
References......Page 72
1 Introduction......Page 74
2 Problem Formulation......Page 76
3.1 Lower Bound......Page 80
3.2 Upper Bound......Page 82
3.3 Special Case......Page 84
4 An Example......Page 85
5 Conclusion......Page 86
References......Page 87
1 Introduction......Page 90
2 One-Shot Optimization and Problem Characteristics......Page 92
3 The Newton Scenario for Separable Adjoints......Page 95
4 Jacobi Method on an Elliptic Problem......Page 99
5 Multigrid Method......Page 104
References......Page 105
1 Introduction......Page 107
2 Generalized Nash Equilibrium Problem......Page 108
3 Equivalent Reformulations......Page 109
4 Evolutionary Algorithm......Page 111
5 Numerical Experiments......Page 113
6 Conclusion......Page 114
Appendix 2: D-stability......Page 116
1 Introduction......Page 117
2 Notations and Preliminaries......Page 119
3.1 The Scalar Optimization Problem (PS)......Page 123
3.2 The Scalar Optimization Problem (PSΣ)......Page 126
3.3 The Vector Optimization Problem (PV)......Page 133
3.4 The Vector Optimization Problem (PVm)......Page 137
References......Page 140
1 Introduction......Page 142
2 Problem Transformation......Page 143
3 A Constant Approximation......Page 144
4 A PTAS......Page 146
References......Page 151
1 Introduction......Page 153
2 Related Work......Page 156
4 Model......Page 158
4.2 Channel Model......Page 159
5 Centralised Power Control......Page 162
6.1.1 The Continuous-Time Algorithm......Page 163
6.1.2 The Discrete-Time Algorithm......Page 164
6.2 Review of Yates\' Framework......Page 165
6.2.2 The Discrete-Time Algorithm......Page 166
7 Power Control in Wireless Networks and Positive Systems......Page 167
8 On the Stability of Power Control Algorithms with Delays......Page 168
9 On Topology Changes in Wireless Networks and Stability......Page 169
10 Illustrative Examples......Page 171
10.1 Power Control with Constant Delays......Page 172
10.2 Power Control with Time-Varying Delays......Page 173
10.3 Quantitative Analysis on the Relation Between the Convergence Rate and Delays......Page 174
10.4 Power Control with Switching Topologies......Page 175
11 Conclusions......Page 178
12 Current Problems and Research Directions......Page 179
Appendix 3: Fixed-points, contractions and para-contractions......Page 181
References......Page 182
1 Introduction......Page 186
2.1 Top-Down Review......Page 187
2.2 Optimization in LUT Models......Page 189
3 Present Applications: Excess Commuting and Sketch Planning......Page 190
3.1 Excess Commuting......Page 191
3.2 Sketch Models......Page 192
3.3 An Example: Calculating a Minimum Energy Urban Layout......Page 194
4 Future Applications and Conclusions......Page 198
References......Page 200
1 Introduction......Page 204
2 Economic Growth with Nonconvex Production Functions......Page 206
3 Numerical Results......Page 210
References......Page 212
1 Introduction......Page 213
2 Survey of General Theory......Page 214
3.1 Gaskins\' Prototype Model of Dynamic Limit Pricing......Page 222
3.2 Cyclical Fluctuations in an Extended Gaskins Model of Dynamic Limit Pricing......Page 224
4 An Interpretation of the Analytical Results......Page 230
References......Page 235
1 Introduction......Page 238
2 Existing Neuro-Fuzzy-Expertsystems......Page 239
2.1 Lin/Lee System......Page 240
2.2 NEFCON System......Page 241
3 An Example......Page 243
5.1.1 Surplus Rules......Page 244
5.1.2 Faulty Rules......Page 245
5.2.2 Creating Rules: Creating Sets......Page 246
5.2.5 Output......Page 247
References......Page 248
1 Introduction......Page 250
2 Two-Point Boundary Value Problem......Page 251
3 Homotopy Method......Page 253
4 Fourth-Order Runge–Kutta Integrator with Switch Function Detection......Page 254
5 Numerical Example......Page 255
5.1 Example 1......Page 256
5.2 Example 2......Page 258
References......Page 263
1 Introduction......Page 264
2 Testing Methods......Page 266
4 Uni-extremum Optimal Control Problem......Page 267
5 Multiextremal Optimal Control Problems......Page 269
6 Optimal Control Problems with Specific Features of Computation......Page 274
7 Bang–Bang OCP......Page 277
8 Conclusion......Page 279
References......Page 280
1 Introduction......Page 282
2 Parallel Computations in the First-Order Methods......Page 284
2.1 Application of Gradient Methods......Page 286
2.2 The Methods for Solving Problem with Constraints on Control......Page 287
2.3 Linearization Method for Solving Problems with Phase Constraints......Page 289
2.4 The Block Scheme of Multimethod Algorithm Operation......Page 290
3.2 Computation of the Method Step......Page 291
3.4 The Methods of Approximation of Control......Page 292
4.1 Example 1......Page 293
5 Conclusion......Page 294
References......Page 295
1 Introduction......Page 296
2 Formulation of a Nonconvex Optimal Control Problem......Page 297
3 Description of Tunneling Algorithm......Page 298
4 The Curvilinear Search Method (The Tunneling Phase)......Page 299
6 Numerical Examples......Page 300
6.1 Test Problem 1......Page 301
6.3 The Test Problem 3......Page 304
References......Page 305
1 Introduction......Page 307
2.1 Interval Arithmetic......Page 308
2.2 The Interval Solution Set Versus the Parametric Solution Set......Page 309
2.3 Prior Work on the Parametric Solution Set......Page 310
3.1 The Residual Iteration Method......Page 311
3.2 Bernstein Enclosure of Polynomial Ranges......Page 312
3.2.1 The Bernstein Form......Page 313
3.2.2 Bernstein Coefficients of Monomials......Page 314
3.2.4 Determination of the Bernstein Enclosure for Polynomials......Page 315
4 Application to Structural Mechanics......Page 316
4.1 One-Bay Steel Frame......Page 317
4.2 Two-Bay Two-Story Frame Model with 13 Parameters......Page 319
4.3 Two-Bay Two-Story Frame Model with 37 Parameters......Page 321
References......Page 322
1 Introduction......Page 325
2 Implicit Runge–Kutta Methods and Stein Matrix Equations......Page 327
3 A Numerical Method for Solving Large Nonsymmetric Stein Matrix Equations......Page 329
3.1 The Extended Block Arnoldi Algorithm......Page 330
3.2 The Extended Block Arnoldi Algorithm for Stein Equations......Page 331
4 Numerical Examples......Page 333
References......Page 335
1 Introduction......Page 337
2 Some Modifications of Chebyshev Method......Page 338
3 Recurrence Relations......Page 339
4 Convergence Study of Chebyshev Method......Page 341
5 Convergence Study of Modifications of the Chebyshev Method......Page 345
6 Mild Convergence Conditions......Page 348
7 Numerical Results and Discussion......Page 349
References......Page 350