Handbook of Optimization Theory: Decision Analysis and Application (Mathematics Research Developments)

Title Page......Page 4
CONTENTS......Page 6
PREFACE......Page 8
Abstract......Page 15
1. General Introduction......Page 16
2.1. Introduction......Page 18
2.2. The Problem Solving Methodology......Page 19
2.2.2. Phase 1: A Guided Local Search Metaheuristic for the TSP......Page 20
2.3.1. The Problem Data and Parameter Setting......Page 22
2.3.2. Analysis of Different Algorithm Configurations......Page 23
3.1. Introduction......Page 27
3.2. Evolution Strategies for Combinatorial Optimization Problems......Page 28
3.3. The Evolution Strategy Algorithm with Multi-Parametric Mutator 
(ES-MPM)......Page 29
3.4. Guided Evolution Strategies (GES) for MGM......Page 30
1. GLS steps......Page 31
3.5. Experiments on MGM Using Simulated Datasets......Page 32
3.6. An Approach to Increase the Map Reliability by Using Verification 
Process......Page 35
4.1. Introduction......Page 36
4.2. Main Idea of the Approach to Solve MCGM......Page 37
4.3. Full Frame (FF) Algorithm for MCGM......Page 38
4.4. Specific Conflicted Frames (SCF) Algorithm for MCGM......Page 39
5.1. Introduction......Page 41
5.1.2. Standard Methodology for Solving Physical Mapping Problem......Page 42
5.2. Reducing PMP to Standard TSP......Page 43
5.2.1. Clustering Algorithm......Page 44
5.2.2. Looking for Linear Topological Structure......Page 45
5.2.4. Re-Sampling Verification of the Obtained Solution......Page 46
5.2.5. Merging of Sub-Contigs......Page 47
Appendix. Choosing Threshold Q Value for PRC Calculation......Page 48
References......Page 49
Abstract......Page 55
1. Introduction......Page 56
2.1. Data Sources......Page 57
2.2. Data Envelopment Analysis......Page 58
2.3. Defining Inputs and Outputs......Page 59
2.4. Sample Selection and Analysis Plan......Page 60
3.1. Scenario A Results......Page 63
3.2. Scenario B Results......Page 68
4. Discussion......Page 73
5. Conclusion......Page 74
References......Page 77
Abstract......Page 81
Introduction......Page 82
Dynamics in a Manpower System......Page 84
Model Assumptions......Page 85
Notations......Page 86
Prediction and Control......Page 87
Problem Description......Page 89
Identification of Homogeneous Groups......Page 90
Model Validation......Page 92
Markov Census Model......Page 93
Markov Models with Known Total Size......Page 94
Proportionality Markov Manpower Models......Page 96
4. Conclusion......Page 97
References......Page 98
Abstract......Page 103
1. Introduction......Page 104
2. Game Formulation with Structural Uncertainties in Payoffs 
and Dynamics......Page 105
3. Characterization of Nash Equilibria......Page 107
4. An Application in Resource Extraction......Page 113
5.1. Game Formulation and Solution Characterization......Page 119
5.2. Infinite Horizon Resource Extraction......Page 123
6. Conclusion......Page 126
Proof of Proposition 4.1.......Page 127
References......Page 129
1. Introduction......Page 131
2. Literature Review......Page 132
3.1. Notation......Page 133
3.2. Problem Formulation......Page 134
3.3. The Solution Procedure......Page 137
VRP-COS......Page 138
4.1. Sub-Problems For Inventory Allocations......Page 139
Sub-IA......Page 140
4.3. A New Solution Scheme......Page 141
5. Numerical Computations......Page 144
References......Page 150
1. Introduction......Page 153
2.1. Hardware Compilation......Page 154
2.2.3. Handel-C Targets......Page 156
3. Iterative Methods......Page 157
3.1. Description of the Jacobi Algorithm......Page 158
5. Experimental Results......Page 159
References......Page 164
1. Introduction......Page 167
2.1. Lyapunov’s Convexity Theorem......Page 169
2.2. Solution Concepts......Page 170
3.1. Characterization of Solutions......Page 173
3.2. Existence of Solutions......Page 174
The Role of Nonatomicity......Page 176
4.1. Monotonicity and Continuity......Page 178
4.2. Closedness of the Lower Partition Range......Page 181
4.3. µ-Concave Functions......Page 183
Convexity of the Lower Partition Range......Page 185
4.4. Submodular Functions......Page 187
Submodularity vs. Supermodularity......Page 189
5.1. Representation by µ-Quasiconcave Functions......Page 190
5.3. Representation by Support Functionals......Page 192
6. Conclusion......Page 194
References......Page 195
1. Introduction......Page 199
2.1. Constitutive Form of Incompressible Hyperelasticity......Page 201
2.2. Variational Formulation......Page 202
2.3. Formulation for Linear Elasticity......Page 203
3.2. Introduction of VOF Function......Page 204
4.1. Design Sensitivity Analysis......Page 206
4.2. Update of VOF Function......Page 208
5.1. Advection Analysis of VOF Function......Page 209
6.1. Minimization of End-Compliance......Page 210
6.3. Topology Optimization of Cantilever......Page 211
6.4. Topology Optimization of Simple Beam......Page 213
7.1. Objective Function and Design Sensitivity......Page 214
7.2. Shape Optimization of Railway Rubber Pad......Page 219
8. Conclusion......Page 223
References......Page 224
1. Introduction......Page 225
1.1. Quasi-even Set Generators of Pareto Frontier......Page 227
2. Multiobjective Optimization Problem. Pareto Optimal Solu-
tion......Page 228
3.1. Trade-off Matrix. Utopia Plane......Page 229
3.2. Modified Anchor Points......Page 230
3.4. Shrinking of Search Domain......Page 231
3.5. Arbitrary Direction of Search Domain......Page 234
3.6. Rotation of Search Domain......Page 235
3.9. Scaling Procedure......Page 237
4.1. Criterion of Evenness......Page 238
5. Pareto Local Approximation......Page 243
6. Example of a Local Approximation......Page 246
7. Conclusion......Page 247
References......Page 248
Abstract......Page 251
1. Introduction......Page 252
2.1. Preliminaries......Page 254
2.2. Mixed Strategies and Coalition Formation Rules......Page 256
Existence......Page 257
Multiplicity......Page 258
3. Evolution of Coherent Coalition Structure Beliefs......Page 259
The viability theorem......Page 260
4. The Regulation Map in the Mixed γ Model......Page 261
The characterization theorem......Page 262
A Paradox in the mixed γ model......Page 266
5. Minimal Change of Beliefs......Page 267
6. Conclusion......Page 270
References......Page 271
1. Introduction......Page 273
2.1. The Perturbed Frozen Time form of a LTV Plant......Page 274
2.2. The Prior Knowledge on the Plant......Page 275
3.1. Continuous Time Systems......Page 277
3.2. Discrete Time Systems......Page 281
4. Numerical Examples on Stability Analysis......Page 284
5.1. Robust Controller Design......Page 293
6. Numerical Examples on Robust Stabilization......Page 295
7. Conclusion......Page 299
References......Page 300
1. Abstract......Page 305
2.1. The Original Problem......Page 306
2.2. Reformulation as a Saddle-Point Problem......Page 307
3.1. Definitions and Preliminary Results......Page 308
3.2. Decomposition of the Convex K......Page 309
3.3. Hierarchical Solution of the Saddle-Point Problem......Page 310
3.3.1. A Subordinated Saddle-Point Problem......Page 312
3.3.2. Relations between Saddle-Points of F and Saddle-Points of FI I......Page 315
3.4. Hierarchical Systems of Variational Inequalities......Page 318
Algorithm......Page 321
4.2. Convergence of the Algorithm......Page 322
5.1.1. Convexity-Concavity......Page 324
5.2. The function FI......Page 325
5.2.1. Convexity-Concavity......Page 327
5.2.2. Continuity......Page 328
5.3.1. Convexity-Concavity......Page 329
5.3.2. Continuity......Page 330
6.1. Differentiability of F......Page 331
6.2. Differentiability of FI......Page 332
6.4. Differentiability of FI I in the Diagonal of KI ×KI......Page 337
6.5. Monotony of the Gradients of FI I......Page 339
References......Page 340
1. Introduction......Page 343
2.1. Production System......Page 345
2.2. Admissible States......Page 346
3.1. Properties of the Optimal Cost Function......Page 347
3.3.2. The HJB Equation......Page 348
4.1. Elements of the Discrete Problem......Page 349
4.1.1. Discretization of HJB Inequalities......Page 351
4.3. Accelerated Algorithm......Page 352
Ergodic Case......Page 353
5.1. Definition of the QVI System and Its Viscosity Solution......Page 354
6.1. Elements of the Discrete Problem......Page 355
6.1.4. Definition of Operator Pα
k......Page 356
Piecewise Deterministic Problem (PWD)......Page 357
7.2. Admissible States......Page 358
7.4. Admissible Controls......Page 359
8.2. The Optimal Cost Function V and Its Properties......Page 360
8.3. A Fixed Point Pproblem and Its Solution......Page 361
8.4. Dynamic Programming Principle......Page 362
9. The Operator P......Page 363
9.1. Operator P Properties......Page 364
9.2. Existence and Uniqueness of Fixed Point......Page 367
9.3. Regularity of the Fixed Point P......Page 370
10.1. Brief Description of the Problem......Page 371
10.2. Properties of the Optimal Cost Function......Page 372
10.3. Approximation of the Singular Problem......Page 375
11.1. Elements of the Discrete Problem......Page 379
11.2.1. A Discrete Fixed Point Problem......Page 380
11.3. Convergence Results......Page 383
12. Applications......Page 387
References......Page 388
1. Introduction......Page 391
2. Generic and Density Results in Vector Optimization......Page 392
3. Vector Optimization with Continuous Objective Functions......Page 393
4. Vector Optimization Problems with Semicontinuous Objec-
tive Functions......Page 394
5. Density Results......Page 396
6. Vector Constrained Optimization......Page 397
References......Page 398
1. Introduction......Page 401
2. Novel Concept......Page 403
3. Preliminary......Page 405
4. Main Idea......Page 408
5. Control Algorithm Using Fuzzy Logic......Page 410
6. Numerical Example......Page 412
7. Dynamic Output Feedback......Page 415
References......Page 417
1. Introduction......Page 421
2. Problem Formulation......Page 423
3. Asymptotic Structure and Local Uniqueness......Page 425
5. Fixed Point Iterations......Page 428
6. Reduced-Order Computation for Solving Stochastic Algeb-
raic Lyapunov Equation......Page 431
7.1. Example 1......Page 433
7.2. Example 2......Page 435
8. Conclusion......Page 436
References......Page 437
1. Introduction......Page 439
1.1. Literature Review......Page 441
1.2.1. Preliminaries about Subprime Mortgages......Page 443
1.2.2. Preliminaries about Subprime Mortgage Models......Page 446
1.2.3. Preliminaries about Subprime Risks......Page 449
2. Risk, Profit and Valuation under RMBSs......Page 452
2.2. Risk and Profit under RMBSs......Page 453
2.2.1. A Subprime Mortgage Model for Risk and Profit under RMBSs......Page 454
2.3. Valuation under RMBSs......Page 455
2.4.1. Statement of Optimal Valuation Problem under RMBSs......Page 456
2.4.2. Solution to Optimal Valuation Problem under RMBSs......Page 457
3.1. Subprime ABS CDOs......Page 459
3.2.1. A Subprime Mortgage Model for Risk and Profit under RMBS CDOs......Page 460
3.2.2. Profit under RMBS CDOs and Retained Earnings......Page 462
3.3. Valuation under RMBS CDOs......Page 463
3.4.1. Statement of Optimal Valuation Problem under RMBS CDOs......Page 464
3.4.2. Solution of Optimal Valuation Problem under RMBS CDOs......Page 465
4. Mortgage Securitization and Capital under Basel Regulation......Page 466
4.2. Subprime Mortgages and Their Rates under Basel Capital Regulation
(Slack Constraint; Securitized Case)......Page 467
4.4.2. Capital Constraint Holding (Securitized Case)......Page 468
5.1.1. Choices of Subprime Mortgage Securitization Parameters......Page 469
5.1.2. Computation of Subprime Mortgage Securitization Parameters......Page 470
5.2.1. Cost of Funds......Page 472
5.2.2. Return on Equity (ROE)......Page 475
5.3. Example of a Subprime RMBS Deal......Page 477
5.4. Comparisons between Two Subprime RMBS Deals......Page 479
5.4.1. Details of AMSI 2005-R2 and SAIL 2006-2......Page 480
5.4.2. Comparisons between AMSI 2005-R2 and SAIL 2006-2......Page 481
6.1. Risk, Profit and Valuation under RMBSs and the SMC......Page 482
6.1.1. Subprime RMBSs and the SMC......Page 483
6.1.2. Risk and Profit under RMBSs and the SMC......Page 484
6.1.4. Optimal Valuation under RMBSs and the SMC......Page 485
6.2.1. Subprime RMBS CDOs and the SMC......Page 486
6.2.2. Risk and Profit under RMBS CDOs and the SMC......Page 488
6.3. Mortgage Securitization and Capital under Basel Regulation and the
SMC......Page 489
6.3.2. Subprime Mortgages and Their Rates under Basel Capital Regulation (Slack
Constraint; Securitized Case) and the SMC......Page 490
6.3.4. Subprime Mortgages and Their Rates under Basel Capital Regulation (Future
Time Periods; Securitized Case) and the SMC......Page 491
6.4.2. Example Involving Profit from Mortgage Securitization and the SMC......Page 492
6.4.3. Example of a Subprime RMBS Deal and the SMC......Page 493
6.4.4. Comparisons between Two Subprime RMBS Deals and the SMC......Page 494
7. Conclusions and Future Directions......Page 495
8.1. Appendix A: Proof of Theorem 2.3......Page 496
8.3. Appendix C: First Order Condition (8.52)......Page 498
8.6. Appendix F: Proof of Theorem 4.1......Page 499
8.8. Appendix H: Proof of Proposition 4.3......Page 501
References......Page 502
Abstract......Page 505
1. Introduction......Page 506
1.1.1. Brief Literature Review of Subprime Mortgage Crisis......Page 508
1.1.2. Brief Literature Review of RMLs and Their Securitization......Page 509
1.1.3. Brief Literature Review of Bank Capital......Page 510
1.2. Preliminaries About the SMC, Securitization, Capital and Profitability......Page 511
1.2.1. Preliminaries about the Subprime Mortgage Crisis......Page 512
1.2.2. Preliminaries about RML Securitization......Page 514
1.2.3. Preliminaries about Profitability......Page 516
1.2.4. Preliminaries about Poisson Processes, Jump Diffusions and L´evy Processes......Page 518
1.2.5. Preliminaries about Shot Noise Processes for Primary & Secondary RML
Losses......Page 520
1.3.1. Main Problems......Page 522
2.1. OR’s General Assets......Page 523
2.1.3. OR’s Intangible Assets......Page 524
2.2.1. RML Demand and Supply......Page 525
2.2.2. RML Pricing......Page 527
2.2.3. Residential Mortgage-Backed Securities (RMBSs)......Page 528
2.2.4. RML Losses and their Provisioning......Page 529
2.3.1. Risk-Weighted Assets......Page 533
2.3.2. OR’s Aggregate Assets......Page 534
2.3.3. OR’s Asset Price Processes......Page 535
2.4.1. OR’s Deposits......Page 536
2.5.1. OR’s Regulatory Capital......Page 537
2.5.2. Stochastic Dynamics of OR’s Capital......Page 538
2.5.3. Stochastic Dynamics of OR’s Equity Capital......Page 539
2.5.4. OR’s Capital Valuation......Page 540
2.6.1. Stochastic Modeling of OR’s NIATBCF......Page 541
2.6.2. Computing OR’s ROA and ROE......Page 544
2.6.3. OR’s Total Nett Profit......Page 545
2.6.4. OR’s Profit Dynamics Under Securitized RMLs......Page 546
3.1. Statement of OR’s Optimal Securitization Problem......Page 547
3.2.2. Solving OR’s Optimal Securitization Problem......Page 548
4.1. Model I: Levy Processes of Jump-Diffusion Type......Page 552
4.2. Model II: An Appropriate Time Series Model......Page 553
4.3. Forecast Evaluation......Page 555
5. RML Securitization, Capital and Profitability and Their
Connections with the Subprime Mortgage Crisis......Page 557
5.1. Problematic RML Subportfolios from Section 2.......Page 558
5.4. Capital-to-Loans Sensitivity as in (2.5) from Subsection 2.2.2.......Page 559
5.6. OR’s Own RML Rate as in (2.6) from Subsection 2.2.2.......Page 560
5.8. Pricing of RMBSs as in (2.7) from Subsection 2.2.3.......Page 561
5.9. RML Losses and Default Rate as in (2.22) from Subsection 2.2.4.......Page 562
5.12. LIBOR as in (2.30) from Subsection 2.4.2.......Page 563
5.14. Contracted Liquidity as in Remark 2.2 from Subsection 2.4.......Page 564
5.16. Profitability as in (2.38) and (2.39) from Subsection 2.6.......Page 565
6. Conclusions and Future Directions......Page 566
7.1. Appendix A1: It o’s Formula for Jump Diffusions......Page 568
7.2. Appendix A2: General Optimal Stopping Problem......Page 569
7.3. Appendix A3: Integro-Variational Inequalities for Optimal Stopping......Page 570
7.5. Appendix B: Definitions......Page 572
References......Page 574
1. Introduction......Page 579
2. Open Queueing Networks......Page 581
3. Closed Queueing Networks......Page 586
Acknowledgements......Page 589
References......Page 590
1. Introduction......Page 591
2.1. Preliminaries......Page 592
2.2. Main Result......Page 594
3. Normwise Condition Number for Underdetermined Linear
System with Full Row-rank Matrix......Page 596
Acknowledgements......Page 598
References......Page 599
1. Introduction......Page 601
1.1. Literature Review......Page 604
1.2.1. Preliminaries about Subprime RML Securitization......Page 605
1.2.2. Preliminaries about Credit Default Swaps......Page 606
1.2.3. Preliminaries about RML Securitization Risks......Page 607
1.3. Main Problems and Outline of the Paper......Page 608
2.1.1. Subprime RMBS Price Process......Page 609
2.1.3. Credit Ratings......Page 610
2.2. IB’s Profit under Optimal Subprime RML Securitization......Page 611
3.1. Statement of the Optimal Credit Default Insurance Problem......Page 612
3.2.1. General Solution to the Optimal Credit Default Insurance Problem......Page 613
3.2.2. Optimal Credit Default Swap Contracts......Page 617
3.2.3. Boundary Value Problem......Page 618
3.2.4. Stochastic Optimal Credit Default Insurance with Exponential Utility......Page 619
3.2.5. Stochastic Optimal Credit Default Insurance with Power Utility......Page 621
3.2.6. Stochastic Optimal Credit Default Insurance with Logarithmic Utility......Page 625
4. Optimal Credit Default Insurance and the Subprime Mortgage Crisis......Page 626
4.1.1. Subprime Securitization Risks and the SMC......Page 627
4.1.3. IB’s Profit under Subprime RML Securitization and the SMC......Page 628
4.1.4. Simulation......Page 629
4.2. Stochastic Optimal Credit Default Insurance and Its Connections with
the SMC......Page 630
4.2.3. Optimal Credit Default Insurance with Exponential Utilities and the SMC......Page 631
4.2.4. Optimal Credit Default Insurance with Power Utilities and the SMC......Page 632
5. Conclusions and Future Directions......Page 633
References......Page 634
INDEX......Page 637