## Global Optimization in Action: Continuous and Lipschitz Optimization: Algorithms, Implementations and ApplicationsIn 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 |

473 | |

### Other editions - View all

Global Optimization in Action: Continuous and Lipschitz Optimization ... János D. Pintér Limited preview - 2013 |

Global Optimization in Action: Continuous and Lipschitz Optimization ... János D. Pintér No preview available - 2010 |

### Common terms and phrases

additional algorithm scheme analysis applied approximation aquifer arbitrary assume assumptions Chapter clustering computational considered constraints convex convex functions convex programming convex set corresponding decision defined denotes detailed deterministic dimensional discrepancy function discrepancy measure discussed environmental equations eutrophication evaluation feasible set finite function f given global optimization problems global search Horst illustrate implies inequality interval LGOP limit point linear Lipschitz Lipschitz constant Lipschitz functions Lipschitz-constant Lipschitz-continuous lower bound method minimal model calibration model form model output model parameters multiextremal nonlinear notation Note objective function observation one-dimensional parameterization partition algorithms partition operator partition strategies partition subsets PAS-type Pinter Piyavskii's pollution problem statement programming random search random variables recall Remark respect robust sample points search phase search points seed point selected sequence solution approaches solve Somlyody statistical Step stochastic model stochastic optimization stochastic programming subinterval Theorem tion univariate vector wastewater treatment water quality