Recent advances in optimization and its applications in engineering

Front Cover......Page 1
Preface......Page 7
Contents......Page 9
Part I Convex Optimization......Page 14
  1 Introduction......Page 16
   Historical remarks......Page 18
   Fractional quadratic problems......Page 19
   Combinatorial problems......Page 20
   Small dimensions......Page 22
   Complete positivity......Page 23
   Copositivity criteria based on structural matrix properties......Page 24
  5 Approximation hierarchies......Page 25
   Optimizing over......Page 27
  Conclusion and outlook......Page 29
  References......Page 30
  1 Introduction......Page 34
  2 Problem Statement......Page 35
  3 Observer Design......Page 37
  4 Case Study: A High Cell Density Bioreactor......Page 39
  5 Conclusion......Page 42
  References......Page 43
  1 Introduction......Page 44
  2 Problem class and examples......Page 45
  3 Finite-dimensional approximations......Page 46
   4.1 Upper bound on......Page 48
   Regularity Theorem......Page 49
   Satisfying the hypotheses of the Regularity Theorem......Page 50
   4.3 Convergence result......Page 51
   X is the Lebesgue space Lq......Page 52
  References......Page 53
  1 Introduction......Page 54
  2 Notation......Page 55
  3 Cones of positive polynomials......Page 56
  4 Sums of squares relaxations......Page 59
  5 Hierarchies of relaxations......Page 61
  References......Page 63
  1 Introduction......Page 64
  2 Problem, algorithm and assumptions......Page 65
  3 Preliminaries......Page 67
  4 Error Bounds......Page 69
  References......Page 73
Part II Nonlinear Optimization......Page 74
 On Hessianand Jacobian-Free SQP Methods a Total Quasi-Newton Scheme with Compact Storage......Page 76
  1 Introduction......Page 77
   2.1 Compact Representation Formula......Page 78
   2.2 Maintaining the Positive Definiteness of the Hessian......Page 79
   2.3 Constrained Optimization and Limited-Memory......Page 80
   2.4 Avoidance of the Null-space Factor......Page 81
   2.5 Improving Computational Efficiency......Page 82
  3 Examples......Page 84
  References......Page 85
  1 Introduction......Page 86
  2 Motivation of the Distance Function......Page 87
  3 Parameterization of the set of ellipsoids......Page 92
  References......Page 95
  1 Introduction......Page 96
  2 Continuous Reformulation......Page 97
  3 Results for MINLP Library......Page 98
  4 Extension of Continuous Reformulation......Page 100
  5 Large-Scale Example from Process Synthesis......Page 103
  References......Page 105
  1 Introduction and Problem Statement......Page 106
  2 Local convergence of SCP methods......Page 108
  3 Numerical Results......Page 113
  References......Page 115
 Fixed-Order H-infinity Optimization of Time-Delay Systems......Page 116
  1 Problem Formulation......Page 117
   2.1 Algorithm......Page 118
   2.2 Computation of the H-infinity Norm......Page 119
   2.3 Prediction Step......Page 120
   2.4 Correction Step......Page 121
   2.5 Computing the Gradients......Page 122
  3 Examples......Page 123
  5 Acknowledgements......Page 124
  References......Page 125
  1 Introduction......Page 126
  2 Minimax Optimization......Page 127
   3.1 Arnoldi Process on First Order System......Page 129
   3.2 SOAR......Page 130
   3.3 PIMTAP......Page 131
   4.1 Computation of Derivatives w.r.t Free Variables......Page 132
  5 Numerical Results......Page 133
  References......Page 134
Part III Optimization on Manifolds......Page 136
  1 Introduction......Page 138
  2 Optimization on manifolds in one picture......Page 139
  3 Why consider general manifolds?......Page 140
  4 Manifolds and smooth objective functions......Page 142
   5.1 Stiefel manifold......Page 144
   5.7 Oblique manifold and products of spheres......Page 145
  6 Steepest descent: from Rn to manifolds......Page 146
   6.1 Tangent vectors and tangent spaces......Page 147
   6.3 Steepest-descent direction and the gradient......Page 148
   6.6 Choice of the search curve......Page 149
  8 Newton’s method on manifolds......Page 150
   8.2 Newton on submanifolds of......Page 151
   8.4 Newton for Rayleigh quotient optimization on unit sphere......Page 152
  Acknowledgements......Page 153
  References......Page 154
 On the Best Low Multilinear Rank Approximation of Higher-order Tensors*......Page 158
  1 Introduction......Page 159
   2.1 Basic definitions......Page 160
   2.2 Best low multilinear rank approximation......Page 161
   2.3 Higher-order singular value decomposition......Page 162
  3 Some applications......Page 163
   4.1 Geometric Newton algorithm......Page 165
   4.2 Trust-region based algorithm......Page 167
   4.3 Conjugate gradient based algorithm......Page 169
   4.4 Remarks......Page 171
  5 Local minima......Page 172
  6 Conclusions......Page 173
  References......Page 174
  1 Introduction......Page 178
  2 Formulation and convex relaxation of sparse PCA......Page 179
  3 A procedure to refine the components......Page 180
  4 Numerical experiments......Page 182
  References......Page 183
  1 Introduction......Page 186
  2 Principal Component Analysis......Page 187
  4 Fitting cosets......Page 188
   4.1 Reduction to a homogeneous space......Page 190
   4.2 An algorithm for Riemannian fitting......Page 191
  5 Example: Fitting on SO(n)......Page 192
   Acknowledgments......Page 194
  References......Page 195
  1 Introduction......Page 196
  2 Mathematical preliminaries......Page 198
  3 The RBFGS Algorithm......Page 199
   4.1 Two Approaches......Page 200
   4.2 Implementation on the Unit Sphere......Page 201
   4.3 Implementation on the Compact Stiefel Manifold St(p,n)......Page 202
  5 Applications and numerical experiment results......Page 203
  References......Page 205
  1 Introduction......Page 206
  2 Our approach......Page 207
  3 Normalization of the coefficient vector......Page 209
  4 Choice of  and the order......Page 213
  5 Conclusions......Page 214
  References......Page 215
Part IV Optimal Control......Page 216
  1 Introduction......Page 218
  2 Two bodies, two controls......Page 222
  3 Two bodies, one control......Page 227
  4 Three bodies, two controls......Page 230
  References......Page 236
  1 Introduction......Page 238
  2 A variational approximation......Page 240
  3 Numerical results......Page 244
   The SPG method......Page 245
   The Euler–Lagrange method......Page 246
   The SPG method......Page 247
  References......Page 248
  2 Path Planning for Container Cranes......Page 250
  3 Bilevel Optimization......Page 251
  4 Bilevel Optimal Control......Page 253
  5 Time Dependent Coupling......Page 255
  6 Main and Alternative Trajectories......Page 256
  References......Page 259
  1 Introduction......Page 260
  2 Basics of the MSE......Page 261
  3 State-constrained arcs......Page 262
   4.1 Full parameterization with prototype adjoints......Page 264
   4.2 Derivatives of cost......Page 265
   4.3 Spike and flat generations of type B procedures......Page 266
   4.4 Partial parameterization with prototype adjoints......Page 267
  References......Page 268
  Appendix......Page 269
  2 Algorithm of MSE......Page 270
  3 State-constrained arcs......Page 272
  4 Full parameterization with prototype adjoints......Page 273
  5 Partial parameterization with prototype adjoints......Page 276
  6 MSE implementation......Page 277
  References......Page 279
  1 Introduction......Page 280
  2 A Mathematical Model for Tumor Anti-Angiogenesis......Page 281
  3 Addition of a Pharmacokinetic Model......Page 284
  4 Suboptimal Approximations......Page 286
  References......Page 288
  1 Introduction......Page 290
  2 Problem formulation......Page 292
  3 Optimality Criteria......Page 293
  4 Duality in Weighted Sobolev Spaces......Page 294
  5 Application to the Prey-Predator model......Page 296
  References......Page 298
Part V Model Predictive Control......Page 300
  1 Introduction......Page 302
  2 Setup and Preliminaries......Page 303
  3 Performance and stability analysis......Page 304
  4 A numerical case study......Page 307
  References......Page 311
  1 Introduction......Page 312
  2 Model Description......Page 314
   3.1 Discrete-time Approximation......Page 315
   4.1 SQP Algorithm......Page 316
   4.2 Gradient Computation......Page 317
   5.2 Objective Function with Soft Output Constraints......Page 318
  6 Optimization Results......Page 319
  7 Conclusion......Page 320
  References......Page 321
  1 Introduction......Page 322
  2 Linear Model Predictive Control......Page 323
   3.2 Primal vs. dual strategies......Page 325
   3.5 Our choice......Page 326
   4.2 The iterative process......Page 327
   4.3 Efficient update method......Page 328
   4.4 Approximate solution & warm start......Page 329
  5 Conclusion......Page 330
  References......Page 331
  1 LQR control with constraints......Page 332
  2 Virtual control regularization......Page 335
  3 Examples......Page 337
   3.1 Inverse Pendulum......Page 338
   3.2 Trolley......Page 339
  References......Page 341
  1 Introduction......Page 342
  2 Setup and Preliminaries......Page 343
  3 Controllability and performance bounds......Page 344
  4 Sampled–data systems and arbitrary fast sampling......Page 345
  5 Growth condition and analytic formula......Page 348
  6 Numerical Examples......Page 350
  References......Page 351
  1 Introduction......Page 352
   1.1 Direct Optimal Control in Nonlinear Model Predictive Control......Page 353
   1.3 Direct Multiple Shooting Discretization......Page 354
   1.4 Sequential Quadratic Programming......Page 355
   2.1 Initial Value Embedding and Tangential Predictors......Page 356
   2.3 Parametric Quadratic Programming......Page 357
   3.2 Mode B: Feasibility Improvement Iterations......Page 358
   3.3 Mode C: Optimality Improvement by Adjoint SQP Iterations......Page 359
   3.5 Assembling Multi-level Iteration Schemes......Page 360
   3.7 Computing the Local Feedback Law......Page 361
   4.1 The Block Structured Quadratic Subproblem......Page 362
   4.2 Condensing and Dense Active Set Methods......Page 363
   4.3 Riccati Recursion......Page 365
   4.4 Block Structured Active Set Methods......Page 366
  References......Page 368
Part VI PDE-Constrained Optimization......Page 372
  2 Application: The Simulated Moving Bed Process......Page 374
  3 Formulation of the Optimization Problem......Page 377
   4.2 Asymptotic Convergence Rates of the Newton-Shift-Picard Method......Page 379
  5 The Newton-Picard Inexact SQP Method......Page 384
   5.2 QP Reduction......Page 385
  6 Simultaneous Approximation of Dominant Subspaces......Page 386
   7.1 Effort per SQP iteration......Page 387
  8 Numerical Results......Page 388
  References......Page 390
 On the Optimization of Steady Bingham Flow in Pipes......Page 392
  1 Regularized problem......Page 393
  2 Optimality system......Page 396
  3 Semi-smooth Newton algorithm and numerical tests......Page 397
  References......Page 399
  1 Introduction......Page 402
  2 Semismooth Newton methods and the primal-dual active set strategy......Page 403
  3 On the state equation......Page 406
  4 Optimal control problem......Page 407
  5 Discretization......Page 408
  6 Numerical examples......Page 409
  References......Page 410
  1 Introduction......Page 412
  2 Optimization of the complex model......Page 413
  3 Space mapping......Page 414
  4 Surrogate optimization......Page 416
  5 Numerical results......Page 418
  References......Page 421
  1 Introduction......Page 422
  2 The monodomain equations......Page 424
   3.1 Optimize before discretize......Page 425
   Discretization of primal and adjoint problems......Page 426
   3.2 Discretize before optimize......Page 427
  4 Results......Page 428
  5 Conclusions......Page 430
  References......Page 431
  1 Hyperthermia Treatment Planning......Page 432
   1.1 The Bio-Heat Transfer Equation......Page 433
   1.2 Inequality constraints and objective......Page 435
  2 Barrier Minimizers and their Optimality Conditions......Page 436
  3 Numerical results......Page 439
  References......Page 441
 On a State-Constrained PDE Optimal Control Problem arising from ODE-PDE Optimal Control......Page 442
  2 The hypersonic rocket car problem......Page 443
  3 The state-unconstrained problem and its associated temperature profile......Page 444
  4 Necessary optimality conditions: Interpretation as state-constrained PDE optimal control problem......Page 446
  5 Numerical results......Page 448
  6 Conclusion......Page 450
  References......Page 451
Part VII Engineering Applications of Optimization......Page 452
  1 Introduction......Page 454
  2 Engineering Challenges for Intelligent Vehicle Systems......Page 455
  3 The optimal design approach......Page 456
  4 Vehicle model......Page 457
  5 Controller modeling......Page 459
  6 Concept model optimization......Page 460
  7 Shock absorber optimization......Page 461
  References......Page 462
  1 Introduction......Page 464
  2 Dynamic control model of a machine tool manipulator......Page 465
  3 Feedback control performance......Page 466
  4 Optimal control models of machine tool manipulators......Page 467
  5 Time-optimal control......Page 468
   5.1 Numerical results......Page 469
   5.2 State constraints......Page 470
  6 Damping-optimal control......Page 471
  7 Conclusion......Page 472
  References......Page 473
  1 Introduction......Page 474
  2 Material distribution formalisms......Page 475
  3 Reproduction mechanisms......Page 476
   3.2 Adapted mechanisms......Page 477
   4.1 Theoretical case......Page 478
  5 Results and discussion......Page 479
 A Variational Model for Image Texture Identification......Page 484
   1.2 The variational model......Page 485
   2.1 Discretization of the problem......Page 486
  3 Numerical tests and comments......Page 489
  4 Conclusion......Page 492
  References......Page 493
  1 Integrated methodology......Page 494
   1.2 Process integration and automation for optimization purpose......Page 495
   Degree of Influence (DOI)......Page 496
  2 Application: mass optimization of a bumper system......Page 497
   2.2 Load cases: reparability low speed impact......Page 498
   3.2 Objectives and constraints......Page 499
   3.5 Bumper Design Optimization......Page 501
  4 Conclusions and discussion......Page 502
  References......Page 503
  1 Introduction......Page 504
   2.1 Learning Automata......Page 506
   2.2 Application to Scheduling......Page 507
   4.1 WSEPT Heuristic versus Learning Automata......Page 508
   4.2 Multi-Stage Scheduling......Page 509
  5 Discussion......Page 510
  References......Page 511
  1 Introduction......Page 512
   2.1 Surrogate-Based Optimization......Page 513
   2.2 Design of Experiments......Page 516
  3 Sample Turbomachinery Design Application......Page 518
  References......Page 521
  1 Introduction......Page 524
   2.1 MRS model......Page 525
   2.2 Model fitting......Page 526
   3.1 Characteristics of MRSI data......Page 527
   3.3 Simultaneous optimization of MRSI signals......Page 528
   3.4 Adaptive alternating minimization......Page 529
  5 Conclusions......Page 530
  References......Page 533
  1 Introduction......Page 534
  2 A model for the dynamical behavior of the melt surfaces......Page 535
  3 Linear stability analysis......Page 538
  4 Minimizing the roughness of the surfaces......Page 541
  5 Conclusions and Outlook......Page 542
  References......Page 543
Author Index......Page 544
Subject Index......Page 546