Preface-Volume 1: Multiparametric Programming .................. xi
List of Authors ................................................ xv
Part I Theory and Algorithms ................................... 1
1 Multiparametric Linear and Quadratic Programming ............ 3
1.1 Introduction ................................................ 3
1.2 Methodology ................................................. 7
1.3 Numerical Examples ......................................... 12
1.3.1 Example 1: Crude Oil Refinery ........................ 12
1.3.2 Example 2: Milk Surplus .............................. 15
1.3.3 Example 3: Model-Based Predictive Control ............ 15
1.4 Computational Complexity ................................... 16
1.5 Concluding Remarks ......................................... 19
Acknowledgments ............................................ 19
Appendix A. Redundancy Check for a Set of Linear Constraints ... 19
Appendix B. Definition of Rest of the Region ................... 21
Literature ................................................. 22
2 Multiparametric Nonlinear Programming ...................... 25
2.1 Introduction ............................................... 25
2.1.1 Motivating Example ................................... 26
2.2 The mp-NLP Algorithm ....................................... 27
2.3 Example .................................................... 32
2.4 Global Optimization Issues ................................. 36
2.4.1 Remarks and Observations on the Application of
the mp-NLP Algorithm for Problem (2.8) ............... 36
2.4.2 Algorithm for Multiparametric Nonlinear
Programming .......................................... 38
2.4.3 Example (2.8) Solved with the New Algorithm .......... 39
2.4.4 Extension to Higher Order Spaces and Higher Order
Objective Functions .................................. 42
2.5 Concluding Remarks ......................................... 43
Appendix A. Infeasibility of Corners ........................... 44
Appendix B. Comparison Procedure ............................... 45
Appendix C. Definition of the Rest of the Region ............... 45
Appendix D. Redundancy Test .................................... 48
Appendix E. Vertices of a Critical Region ...................... 50
Acknowledgments ............................................ 51
Literature ................................................. 51
3 Multiparametric Mixed-Integer Linear Programming ........... 53
3.1 Parametric Mixed-Integer Linear Programming ................ 53
3.2 Multiparametric Mixed-Integer Linear Programming.
Branch and Bound Approach .................................. 56
3.3 Multiparametric Mixed-Integer Linear Programming.
Parametric and Integer Cuts ................................ 62
3.3.1 Initialization ....................................... 63
3.3.2 Multiparametric LP Subproblem ........................ 63
3.3.3 MILP Subproblem ...................................... 64
3.3.4 Comparison of Parametric Solutions ................... 65
3.3.5 Multiparametric MILP Algorithm ....................... 65
3.4 Numerical Example .......................................... 67
3.5 Concluding Remarks ......................................... 68
Appendix A. Definition of an Infcasiblc Region ................. 69
Literature ................................................. 71
4 Multiparametric Mixed-Integer Quadratic and Nonlinear
Programming ................................................ 73
4.1 Introduction ............................................... 73
4.2 Methodology ................................................ 74
4.3 The mp-MIQP Algorithm ...................................... 75
4.3.1 Initialization ....................................... 75
4.3.2 Primal Subproblem .................................... 76
4.3.3 Master Subproblem .................................... 76
4.3.4 Strategy for the Solution of the Master Subproblem ... 77
4.3.5 Envelope of Solutions ................................ 78
4.3.6 Redundant Profiles ................................... 78
4.4 The mp-MINLP Algorithm ..................................... 80
4.4.1 Initialization ....................................... 81
4.4.2 Primal Subproblem .................................... 81
4.4.3 Master Subproblem .................................... 82
4.4.4 Remarks and Summary of the Algorithm ................. 83
4.5 Examples ................................................... 85
4.5.1 Example on mp-MIQP ................................... 85
4.5.2 Example on mp-MINLP .................................. 90
4.6 Concluding Remarks ......................................... 95
Acknowledgment ............................................. 96
Literature ................................................. 96
5 Parametric Global Optimization ............................. 99
5.1 Introduction ............................................... 99
5.2 Parametric Global Optimization ............................ 100
5.2.1 B&B Algorithm ....................................... 101
5.2.2 Multiparametric Convex Nonlinear Programs ........... 102
5.3 Multiparametric Nonconvex Nonlinear Programming ........... 103
5.3.1 Motivating Examples ................................. 103
5.3.2 An Algorithm for Multiparametric Nonconvex
Nonlinear Programming ............................... 116
5.4 Multiparametric Mixed-Integer Nonconvex Programming ....... 118
5.5 Numerical Examples ........................................ 120
5.5.1 Example 1 ........................................... 120
5.5.2 Example 2 ........................................... 120
5.6 Concluding Remarks ........................................ 124
Acknowledgments ........................................... 124
Appendix A. Comparison of Parametric Solutions ................ 124
Appendix B. Definition of Rest of the Region .................. 125
Literature ................................................ 125
6 Bilevel and Multilevel Programming ........................ 129
6.1 Introduction .............................................. 129
6.1.1 Global Optimum of a Bilevel Programming Problem ..... 131
6.2 Quadratic Bilevel Programming ............................. 132
6.2.1 LP|LP Bilevel Programming Problem ................... 135
6.2.2 LP|QP Bilevel Programming Problem ................... 137
6.2.3 QP|QP Bilevel Programming Problem ................... 138
6.3 Bilevel Programming with Uncertainty ...................... 138
6.4 Mixed-Integer Bilevel Programming ......................... 140
6.5 Other Multilevel Optimization Problems .................... 142
6.5.1 Three-Level Programming Problem ..................... 144
6.5.2 Bilevel Multifollower Programming Problem ........... 145
6.6 Concluding Remarks ........................................ 146
Acknowledgments ........................................... 146
Appendix A. ................................................... 146
Literature ................................................ 146
7 Dynamic Programming ....................................... 151
7.1 Introduction .............................................. 152
7.2 Constrained Dynamic Programming ........................... 156
7.3 Illustrative Examples ..................................... 161
7.4 Complexity Analysis ....................................... 168
7.5 Concluding Remarks ........................................ 172
Acknowledgments ........................................... 171
Literature ................................................ 172
Part II Applications ......................................... 173
8 Flexibility Analysis via Parametric Programming .......... 175
8.1 Introduction ............................................. 175
8.2 Flexibility Test and Index for Linear Systems ............ 175
8.2.1 Parametric Programming Approach .................... 175
8.2.2 Algorithm 8.1 ...................................... 177
8.2.3 Illustrative Example ............................... 178
8.2.4 Remarks on Algorithm 8.1 ........................... 181
8.2.5 Design Optimization of Linear Systems .............. 184
8.3 Stochastic Flexibility of Linear Systems ................. 186
8.3.1 Parametric Programming Approach .................... 186
8.3.2 Algorithm 8.2 ...................................... 187
8.3.3 Illustrative Example ............................... 188
8.3.4 Remarks on Algorithm 8.2 ........................... 189
8.4 Expected Stochastic Flexibility of Linear Systems ........ 192
8.5 Process Example 8.1: Chemical Complex .................... 192
8.5.1 Flexibility Test and Index ......................... 193
8.5.2 Design with Optimal Degree of Flexibility .......... 194
8.5.3 Expected Stochastic Flexibility .................... 195
8.6 Process Example 8.2: HEN with 2 Hot, 2 Cold Streams ...... 197
8.6.1 Flexibility Test and Index ......................... 198
8.6.2 Stochastic Flexibility ............................. 199
8.7 Process Example 8.3: HEN with 4 Hot, 3 Cold Streams ...... 200
8.7.1 Flexibility Test and Index ......................... 200
8.7.2 Stochastic Flexibility ............................. 202
8.8 Incorporation of Discrete Decisions ...................... 204
8.9 Extension to Multipurpose Processes ...................... 206
8.10 Flexibility Test and Index for Convex Nonlinear
Systems .................................................. 206
8.10.1 Parametric Programming Approach ................... 206
8.10.2 Algorithm 8.3 ..................................... 207
8.10.3 Illustrative Example .............................. 208
8.10.4 Remarks on Algorithm 8.3 .......................... 211
8.11 Design Optimization of Nonlinear Convex Systems .......... 214
8.12 Stochastic Flexibility of Nonlinear Convex Systems ....... 216
8.12.1 Algorithm 8.4 ..................................... 216
8.12.2 Illustrative Example .............................. 218
8.13 Flexibility Test and Index for Nonlinear Nonconvex
Systems .................................................. 220
8.13.1 Parametric Programming Approach ................... 220
8.13.2 Algorithm 8.5 ..................................... 221
8.13.3 Process Example 8.4 ............................... 221
8.14 Summary and Conclusions .................................. 223
Literature .................................................... 227
9 Planning and Material Design Under Uncertainty ............ 229
9.1 Introduction .............................................. 229
9.2 Process Planning Under Uncertainty ........................ 232
9.3 Supply Chain Planning Under Uncertainty ................... 235
9.4 Hierarchical Decision Planning ............................ 242
9.5 Material Design Under Uncertainty ......................... 246
9.5.1 Material Design Example ............................. 248
9.6 Concluding Remarks ........................................ 252
Acknowledgments ............................................... 252
Literature .................................................... 253
10 Multiobjective Energy and Environmental Analysis ......... 255
10.1 Introduction ............................................. 255
10.2 Review of Hydrogen Infrastructure Studies ................ 256
10.3 Motivation ............................................... 260
10.4 Methodology and Model Overview ........................... 261
10.5 Model Formulation ........................................ 269
10.5.1 Centralized Production Sites and Technologies ..... 269
10.5.2 Distribution Network .............................. 272
10.5.3 Forecourt Markets ................................. 273
10.5.4 Net Present Value Objective Function .............. 275
10.5.5 Greenhouse Gas Emissions Objective Function ....... 279
10.5.6 Model Summary ..................................... 280
10.6 Solution Method .......................................... 281
10.6.1 Model Decomposition Algorithm ..................... 281
10.6.2 Solution Time Comparison .......................... 284
10.7 Illustrative Example ..................................... 285
10.7.1 Problem Formulation ............................... 285
10.7.2 Trade-Off Analysis Results ........................ 288
10.7.3 Constrained Roadmap Comparisons ................... 297
10.8 Conclusions .............................................. 304
Literature ............................................... 304
Index .................................................... 307
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