# A Set of Examples of Global and Discrete Optimization: Applications of Bayesian Heuristic Approach

Springer Science & Business Media, Jul 31, 2000 - Business & Economics - 321 pages
This book shows how the Bayesian Approach (BA) improves well known heuristics by randomizing and optimizing their parameters. That is the Bayesian Heuristic Approach (BHA). The ten in-depth examples are designed to teach Operations Research using Internet. Each example is a simple representation of some impor tant family of real-life problems. The accompanying software can be run by remote Internet users. The supporting web-sites include software for Java, C++, and other lan guages. A theoretical setting is described in which one can discuss a Bayesian adaptive choice of heuristics for discrete and global optimization prob lems. The techniques are evaluated in the spirit of the average rather than the worst case analysis. In this context, "heuristics" are understood to be an expert opinion defining how to solve a family of problems of dis crete or global optimization. The term "Bayesian Heuristic Approach" means that one defines a set of heuristics and fixes some prior distribu tion on the results obtained. By applying BHA one is looking for the heuristic that reduces the average deviation from the global optimum. The theoretical discussions serve as an introduction to examples that are the main part of the book. All the examples are interconnected. Dif ferent examples illustrate different points of the general subject. How ever, one can consider each example separately, too.

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### Contents

 GENERAL IDEAS OF BAYESIAN HEURISTIC APPROACH 3 2 DIRECT BAYESIAN APPROACH DBA 4 3 BAYESIAN HEURISTIC APPROACH BHA 5 4 ILLUSTRATIVE EXAMPLES 6 5 HEURISTICS 7 EXPLAINING BAYESIAN HEURISTIC APPROACH BY EXAMPLE OF KNAPSACK PROBLEM 11 1 EXACT ALGORITHMS 12 2 APPROXIMATE ALGORITHMS 13
 42 MATRIX GAME ALGORITHM 157 5 ONEDIMENSIONAL EXAMPLE 158 6 ECONOMIC DUEL NASH MODEL 159 62 DYNAMIC NASH MODEL 161 7 SOFTWARE EXAMPLE 163 71 OPERATING DUEL 165 FUTURE DEVELOPMENTS 169 PORTFOLIO PROBLEM OPTIMAL INVESTMENT OF RESOURCES 173

 22 GREEDY HEURISTICS 14 24 PERMUTATION 16 3 SOFTWARE EXAMPLES OF KNAPSACK PROBLEM 21 31 C++ 22 32 JAVA 23 SOFTWARE FOR GLOBAL OPTIMIZATION 31 INTRODUCTION TO SOFTWARE 33 12 DESCRIPTION OF METHODS 34 13 APPLICATION AREAS 36 15 COMPUTING ENVIRONMENT 37 3 DIFFERENT VERSIONS 39 33 UNIX C++ INTERACTIVE 40 4 WEBSITES 41 PORTABLE FORTRAN VERSION GMF 45 2 GENERAL DISCUSSION 47 22 LIST OF METHODS 48 3 PROGRAM DESCRIPTION 49 33 MAIN PROGRAM 50 34 EXAMPLE OF THE MAIN PROGRAM 51 TURBO C VERSION TCGM 55 2 USERS REFERENCE 56 24 MINIMIZATION 57 26 NAVIGATION 60 27 MOVING 61 C++ VERSION GMC 63 2 USERS REFERENCE 64 24 MENU SYSTEM 65 JAVA JDK10 VERSION GMJO 75 2 RUNNING GMJO 76 23 GRAPHICAL INTERFACE 77 24 IMPLEMENTING NEW METHOD 80 JAVA JDK11 AND JDK12 VERSIONS GMJ1 AND GMJ2 85 2 STARTING GMJ 86 22 CONFIGURING STANDALONE APPLICATION 88 32 TASK SELECTION 90 33 OPERATION CONTROL 92 4 UPDATING GMJ 93 41 NEW TASKS 95 42 NEW METHODS 97 43 CONFIGURING PROPERTY MANAGER 98 44 NEW ANALYSIS OBJECTS 100 45 PROGRAMMING TIPS 103 46 SECURITY RESTRICTIONS 104 5 FEATURES OF GMJ2 105 52 DOMAIN CONSTRAINT FUNCTION 106 53 RUNNING GMJ2 107 EXAMPLES OF MODELS 113 COMPETITION MODEL WITH FIXED RESOURCE PRICES NASH EQUILIBRIUM 115 2 NASH MODEL 116 3 SEARCH FOR NASH EQUILIBRIUM 118 31 EXISTENCE OF NASH EQUILIBRIUM 119 41 EXAMPLE OF SERVER COALITION 120 COMPETITION MODEL WITH FREE RESOURCE PRICES WALRAS EQUILIBRIUM 123 2 SEARCH FOR WALRAS EQUILIBRIUM 126 3 MONTECARLO SIMULATION 128 32 TESTING EQUILIBRIUM CONDITIONS 131 4 SOFTWARE EXAMPLE 136 INSPECTION MODEL 143 2 SEARCH FOR EQUILIBRIUM 144 21 DIRECT SEARCH ALGORITHM DSA 145 22 NECESSARY AND SUFFICIENT CONDITIONS 146 24 STRATEGY ELIMINATION ALGORITHM SEA 148 DUEL PROBLEM DIFFERENTIAL GAME MODEL 151 2 CONVEX VERSION 152 3 MIXED STRATEGIES 153 4 SEARCH FOR EQUILIBRIUM 155
 2 EXPECTED UTILITY 174 3 OPTIMAL PORTFOLIO SPECIAL CASES 175 4 UTILITY FUNCTIONS 176 5 SOFTWARE EXAMPLE 178 52 A SET OF UTILITY FUNCTIONS 179 53 PREDICTING INVESTMENT RESULTS 180 54 DATA 182 55 RESULTS 183 56 FUTURE DEVELOPMENTS 184 EXCHANGE RATE PREDICTION TIME SERIES MODEL 187 2 AUTO REGRESSIVE MOVINGAVERAGE MODELS ARMA 189 31 OPTIMIZATION OF AR PARAMETERS 190 32 OPTIMIZATION OF MA PARAMETERS 191 34 EVALUATION OF ARMA PREDICTION ERRORS 192 4 EXTERNAL FACTORS 193 41 MISSING DATA 196 6 ARTIFICIAL NEURAL NETWORKS MODELS ANN 197 7 BILINEAR MODELS BL 199 82 MINIMIZATION OF RESIDUALS 200 83 DISCUSSIONS 202 9 MULTISTEP PREDICTIONS 204 10 STRUCTURAL STABILIZATION 205 102 SIMPLE EXAMPLE 208 103 EXAMPLES OF STRUCTURAL OPTIMIZATION WITH EXTERNAL FACTORS 209 11 EXAMPLES OF SQUARED RESIDUALS MINIMIZATION 210 112 OPTIMIZATION RESULTS 215 12 SOFTWARE EXAMPLES 221 122 JAVA VERSION OF ARMA SOFTWARE ARMAJ 238 123 ANN SOFTWARE 243 CALL CENTER MODEL 245 12 ASSUMPTIONS NOTATIONS AND OBJECTIVES 246 2 CALCULATION OF STATIONARY PROBABILITIES 247 3 ASYMPTOTIC EXPRESSIONS 248 5 CALL RATE ESTIMATE 249 7 MONTE CARLO SIMULATION MCS 250 72 MONTE CARLO ERRORS 252 73 STOPPING MONTE CARLO 253 MONTE CARLO MODEL 254 9 TIMEDEPENDANT CASES 255 91 SIMPLE EXAMPLE 256 10 CALL RATE PREDICTIONS 257 102 CALL RATE PREDICTION BY SCALE MODELS 259 103 EXPERT MODEL EVENT SCALE VERSION 261 104 TIME SCALE VERSION VECTOR PREDICTION 269 105 TIME SERIES MODEL ARMA 272 106 APPLICATION EXAMPLES 273 OPTIMAL SCHEDULING 275 2 FLOWSHOP PROBLEM 276 22 HEURISTICS 277 24 GMC SOFTWARE EXAMPLE 278 3 SCHOOL SCHEDULING 282 31 CONSTRAINTS 283 33 PERMUTATION AND EVALUATION ALGORITHM 284 34 SOFTWARE EXAMPLE 285 SEQUENTIAL STATISTICAL DECISIONS MODEL BRIDE PROBLEM 291 2 AVERAGE UTILITY 292 3 SINGLEMARRIAGE CASE 293 32 DISCRETE APPROXIMATION 294 33 INCLUDING THE WAITING COST 295 4 MULTIMARRIAGE CASE 296 42 BELLMANS EQUATIONS 297 5 SOFTWARE EXAMPLES 299 References 307 Index 317 Copyright