Preface ....................................................... vii
I OPTIMIZATION MODELS ........................................... 1
1 ONE VARIABLE OPTIMIZATION .................................... 3
1.1 The five-step Method .................................... 3
1.2 Sensitivity Analysis .................................... 9
1.3 Sensitivity and Robustness ............................. 14
1.4 Exercises .............................................. 16
2 MULTIVARIABLE OPTIMIZATION .................................. 21
2.1 Unconstrained Optimization ............................. 21
2.2 Lagrange Multipliers ................................... 31
2.3 Sensitivity Analysis and Shadow Prices ................. 41
2.4 Exercises .............................................. 50
3 COMPUTATIONAL METHODS FOR OPTIMIZATION ...................... 57
3.1 One Variable Optimization .............................. 57
3.2 Multivariable Optimization ............................. 66
3.3 Linear Programming ..................................... 74
3.4 Discrete Optimization .................................. 91
3.5 Exercises ............................................. 102
II DYNAMIC MODELS ............................................. 113
4 INTRODUCTION TO DYNAMIC MODELS ............................. 115
4.1 Steady State Analysis ................................. 115
4.2 Dynamical Systems ..................................... 120
4.3 Discrete Time Dynamical Systems ....................... 126
4.4 Exercises ............................................. 132
5 ANALYSIS OF DYNAMIC MODELS ................................. 139
5.1 Eigenvalue Methods .................................... 139
5.2 Eigenvalue Methods for Discrete Systems ............... 144
5.3 Phase Portraits ....................................... 150
5.4 Exercises ............................................. 164
6 SIMULATION OF DYNAMIC MODELS ............................... 171
6.1 Introduction to Simulation ............................ 171
6.2 Continuous-Time Models ................................ 178
6.3 The Euler Method ...................................... 186
6.4 Chaos and Fractals .................................... 191
6.5 Exercises ............................................. 206
III PROBABILITY MODELS ........................................ 221
7 INTRODUCTION TO PROBABILITY MODELS ......................... 223
7.1 Discrete Probability Models ........................... 223
7.2 Continuous Probability Models ......................... 228
7.3 Introduction to Statistics ............................ 231
7.4 Diffusion ............................................. 236
7.5 Exercises ............................................. 241
8 STOCHASTIC MODELS .......................................... 251
8.1 Markov Chains ......................................... 251
8.2 Markov Processes ...................................... 261
8.3 Linear Regression ..................................... 271
8.4 Time Series ........................................... 280
8.5 Exercises ............................................. 290
9 SIMULATION OF PROBABILITY MODELS ........................... 301
9.1 Monte Carlo Simulation ................................ 301
9.2 The Markov Property ................................... 308
9.3 Analytic Simulation ................................... 317
9.4 Particle Tracking ..................................... 323
9.5 Fractional Diffusion .................................. 335
9.6 Exercises ............................................. 347
Afterword ..................................................... 359
Index ......................................................... 363
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