1 GLOBAL OPTIMIZATION 6.0 GLOBAL NONLINEAR OPTIMIZATION USING MATHEMATICA Loehle Enterprises 1258 Windemere Ave. Naperville, IL 60564 USA 630-527-8402 630-416-9902 fax [email protected] www.loehleenterprises.com Version 6.0 (C) Copyright 1998-2006 Loehle Enterprises. No part of this manual or software may be copied or reverse engineered except that purchasers of this package may make back up copies of the software. ***************************************************************** DISCLAIMER ***************************************************************** NOTICE: The software product described in this manual is a research tool. No warranty is offered as to operability of this product. No guarantee can be made that any particular nonlinear equation can be solved. No safety critical or financial decisions should be based on the operation of this software. The manufacturer accepts no liability for losses resulting from its use. ***************************************************************** 2 Contents I GETTING STARTED .........................................................................................5 I.1 Welcome ...........................................................................................................................5 I.2 Registering .................................................................................................5 I.3 Installation ..................................................................................................5 I.3.A Recommended System Capabilities ....................................................5 I.3.B Installation on a Hard Drive ...............................................................5 I.4 A Quick Start and Example .......................................................................6 II TIPS FOR PERFORMANCE ........................................................................11 II.1 Improving Performance ............................................................................11 II.2 The Compile Function ..............................................................................12 II.3 User Functions: Working with Named Functions ....................................13 III GENERAL NONLINEAR OPTIMIZATION: GlobalSearch, GlobalPenaltyFn, IntervalMin AND MultiStartMin ..........................................................................14 III.1 Introduction .............................................................................................14 III.2 Example ...................................................................................................16 III.3 Program Operation ..................................................................................16 III.3.A Parameters and Default Values ........................................................16 III.3.B Bounds ..........................................................................................17 III.3.C Constraints .....................................................................................17 III.4 Output ......................................................................................................23 III.5 Maximization ..........................................................................................24 III.6 Limitations ..............................................................................................24 III.7 Error Messages ........................................................................................26 III.8 Performance/Speed ..................................................................................28 III.9 Testing and Examples .............................................................................30 IV THE GlobalMinima FUNCTION .................................................................38 IV.1 Introduction .............................................................................................38 IV.2 The Grid Refinement Approach to Nonlinear Optimization ..................38 IV.3 Program Operation ..................................................................................42 IV.3.A Parameters and Default Values ........................................................42 IV.3.B Bounds ..........................................................................................43 IV.3.C Constraints ....................................................................................43 IV.3.D Memory Usage ..............................................................................44 IV.4 Output ......................................................................................................44 3 IV.5 Maximization ..........................................................................................44 IV.6 Limitations ..............................................................................................45 IV.7 Efficiency ................................................................................................46 IV.8 Error Messages ........................................................................................47 IV.9 A STEP-BY-STEP EXAMPLE ..............................................................48 IV.10 Testing ...................................................................................................52 IV.10.A A Simple Polynomial ....................................................................53 IV.10.B The Rosenbrock Function ..............................................................54 IV.10.C The Branin rcos Function ..............................................................55 IV.10.D The Csendes Function ................................................................................................................57 IV.10.E Wavy Functions, the W Function ...................................................60 IV.10.F More Wavy Functions .................................................................61 IV.10.G Wavy Fractals ............................................................................63 V NONLINEAR REGRESSION: THE NLRegression FUNCTION ...........................................................................................................................................64 V.1 Introduction ..............................................................................................64 V.2 Utilizing Chi-Square fit Criteria ...............................................................71 V.3 Multiple Independent Variables in Regression Problems ........................71 VI MAXIMUM LIKELIHOOD ESTIMATION: THE MaxLikelihood FUNCTION ...........................................................................................................................................78 VI.1 Introduction .............................................................................................78 VI.2 Examples and Built-In Functions ............................................................78 VII DISCRETE VARIABLE PROBLEMS: THE InterchangeMethodMin & TabuSearchMin FUNCTIONS ...................85 V.1I Introduction .............................................................................................85 VII.2 Applications ...........................................................................................85 VII.2.A Capital Allocation ................................................................................................................85 VII.2.B Vehicle Routing/Travelling Salesman ................................................................................................................86 VII.2.C Minimum Spanning Tree ................................................................................................................89 VIII THE MaxAllocation FUNCTION .............................................................90 VIII.1 Introduction ..........................................................................................90 VIII.2 Applications ..........................................................................................92 VIII.2.A Investment Allocation Problems ................................................................................................................93 VIII.2.B Derivative Hedging with Quadratic Programming ................................................................................................................93 4 IX APPLICATIONS ................................................................................................94 IX.1 Zeros of a Function/Roots of a Polynomial .................................................94 IX.2 Integer Programming: the Knapsack Problem ........................................96 IX.3 Differential Equation Models ..................................................................98 IX.4 Constraint Equations ...............................................................................100 X. Literature Cited ..............................................................................................102 5 I GETTING STARTED I.1 Welcome Welcome to Global Optimization. This package provides a suite of tools for solving nonlinear optimization prob- lems, as well as a variety of other applications such as finding the roots or zeros of a nonanalytic function. The package is easier, simplier, and more robust than most optimization tools, and you will find yourself working more efficiently and more easily than before. I.2 Registering Before you continue please register by emailing your name and address with a notice that you have purchased Global Optimization to [email protected]. This will enable us to reach