Global Optimization in Action: Continuous and Lipschitz Optimization: Algorithms, Implementations and Applications

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Springer Science & Business Media, Nov 30, 1995 - Mathematics - 480 pages
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In science, engineering and economics, decision problems are frequently modelled by optimizing the value of a (primary) objective function under stated feasibility constraints. In many cases of practical relevance, the optimization problem structure does not warrant the global optimality of local solutions; hence, it is natural to search for the globally best solution(s).
Global Optimization in Action provides a comprehensive discussion of adaptive partition strategies to solve global optimization problems under very general structural requirements. A unified approach to numerous known algorithms makes possible straightforward generalizations and extensions, leading to efficient computer-based implementations. A considerable part of the book is devoted to applications, including some generic problems from numerical analysis, and several case studies in environmental systems analysis and management. The book is essentially self-contained and is based on the author's research, in cooperation (on applications) with a number of colleagues.
Audience: Professors, students, researchers and other professionals in the fields of operations research, management science, industrial and applied mathematics, computer science, engineering, economics and the environmental sciences.
 

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Contents

General Problem Statement and Special Model Forms
3
112 Concave Minimization
6
113 Differential Convex DC Programming
10
114 Lipschitz Global Optimization
14
Solution Approaches
21
121 Global versus Local Optimization
22
122 Globalized Local Optimization Strategies Combined with Grid Search or Random Search
24
123 Sequential Improvement of Local Minima
27
415 Lipschitzian Equations and Inequalities
258
416 Concluding Remarks
259
Data Classification Clustering and Related Problems
261
Problem Statement and Examples
263
423 Classification Procedures Based on Cluster Seed Points
266
424 Selecting SP M Applying Global Optimization
269
425 Numerical Examples
270
426 Concluding Remarks
274

124 Enumeration of all Minima
29
125 Relaxation Successive Outer Approximation Strategies
31
126 BranchandBound Strategies
33
127 Concluding Remarks
37
Partition Strategies in Global Optimization The Continuous and the Lipschitzian Case
39
An Introduction to Partition Algorithms
41
A General Scheme
46
213 Uniform Grid Search
48
214 Piyavskiis Algorithm
51
215 Kushners Algorithm
56
Convergence Properties of Adaptive Partition Algorithms
59
222 Sufficient and Necessary Convergence Conditions
64
Partition Algorithms on Intervals
75
232 An Efficiency Estimate
85
Partition Algorithms on Multidimensional Intervals
91
242 Convergence of Multivariate Rectangular Partition Methods
97
243 An Efficiency Estimate
100
244 Decomposable Partition Operators
104
Simplex Partition Strategies
111
252 Convergence of Simplicial Algorithms
115
Partition Methods on General Convex and Star Sets
119
262 Lipschitzian Extension of the Objective Function
120
263 Linearly Constrained Feasible Sets
125
264 Nonlinearly Constrained Convex Sets
127
265 General StarShaped Sets
129
Partition Strategies in General Lipschitz Optimization
131
272 A BranchandBound Algorithm Scheme for Lipschitzian Optimization
132
273 Convergence
139
Implementation Aspects Algorithm Modifications and Stochastic Extensions
145
Diagonally Extended Univariate Algorithms for Multidimensional Global Optimization
147
312 Diagonally Extended Univariate Algorithms
148
313 Examples
151
Estimation of Lipschitzian Problem Characteristics in Global Optimization
155
322 SubsetSpecific Estimates of Lipschitzian Characteristics
157
323 Bounding Procedures on the Basis of Sample Points
158
324 LipschitzConstant Estimation Using Results from Extreme Order Statistics
160
325 Numerical Comments and Conclusions
165
General Lipschitz Optimization Applying Penalty Multipliers
169
332 Solution Approach
170
An Implementation of a Lipschitzian Global Optimization Procedure
173
342 Current System Requirements and Problem Size Limitations Hardware
177
343 Using LGO in an Interactive Environment
179
344 Illustrative Test Results
183
Decision Making under Uncertainty Stochastic Model Forms
191
352 Model Variants and Solution Approaches
193
353 Conclusions
202
Adaptive Stochastic Optimization Procedures
205
362 Convergence of Random Search Based Stochastic Optimization Methods
207
363 Convergence of Stochastically Combined Optimization Procedures
216
Estimation of NoisePerturbed Function Values
227
372 Estimation of Noisy Function Values
228
373 Estimation of Probabilities
232
Applications
237
Introductory Notes
239
Nonlinear Approximation Systems of Equations and Inequalities
241
412 Systems of Nonlinear Equations
243
413 Nonlinear Equations and Equivalent Global Optimization Problems
244
414 Test Results on Randomly Generated System of Equations
249
Aggregation of Negotiated Expert Opinions
277
432 A General Model for Combining Negotiated Expert Opinions
279
433 Model Specifications
281
434 Solution Approach Based on Lipschitz Global Optimization
285
435 Numerical Examples
287
436 Conclusions
293
Product Mixture Design
295
442 Solution Approach
298
443 Calculation of Lower Bounds
299
444 Numerical Examples and Remarks
300
Globally Optimized Calibration of Complex System Models
303
A General Problem Statement
306
453 Basic Underlying Assumptions
308
454 Discretization
309
456 Multiple Calibration Objectives
312
457 Multiextremality
314
Calibration Model Versions Illustrated by Examples
317
462 Average lpDistance and Variants
318
463 Discrepancy Measures for Calibrating Soft Systems
321
464 A SedimentWater Interaction Model for Shallow Lakes
328
465 A Chemical Fate Model
331
466 A River Flow Model
334
Dynamic Modelling of Phosphorus Release from Sediments
341
472 Numerical Results and Discussion
343
473 Conclusions
352
Aquifer Model Calibration
353
482 Study Area
355
483 Aquifer Model
356
484 Selection of Optimized Parameters
357
485 Solution Approach
358
487 Conclusions
360
Industrial Wastewater Management
361
492 EnvironmentEconomy Integration in Modelling
362
493 Wastewater Treatment Engineering System Model
363
494 Analytical Optimization Model
370
495 Solution Approaches
372
496 Implementation Aspects
376
497 Illustrative Numerical Results and Discussion
377
Multiple Source River Pollution Management
383
4102 Modelling
384
4103 Solution Method
388
4104 Illustrative Results and Discussion
389
Lake Eutrophication Management
395
Management Alternatives
396
4113 Decomposition and Aggregation
397
4114 A Stochastic Modelling Framework
400
4115 Solution Method
403
Risk Management of Accidental Water Pollution
407
Principles Models Solution Methods
408
4123 An Illustrative Case Study
411
4124 Quantitative Analysis
413
4125 Numerical Example and Discussion
420
Afterword
423
Some Further Research Perspectives
425
References
431
Index
473
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