Optimization Toolbox For Use with MATLAB Computation Visualization Programming User's Guide The MathWorks Version 2
Computation Visualization Programming For Use with MATLAB® User’s Guide Version 2 Optimization Toolbox
Contents Preface Using This Guide Related products Typographical Conventions Introduction What Is the Optimization To 1-2 New Features in version 2.2 ..13 New fsolve Default algorithm Configuration Information Technical conventions 1-5 Matrix Vector and Scalar notation Acknowledgment Tutorial Introduction Problems Covered by the Toolbox Using the Optimization Functions 25
i Contents Preface Using This Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Related Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Typographical Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi 1 Introduction What Is the Optimization Toolbox? . . . . . . . . . . . . . . . . . . . . . 1-2 New Features in Version 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-3 New fsolve Default Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 1-3 Configuration Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-4 Technical Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-5 Matrix, Vector, and Scalar Notation . . . . . . . . . . . . . . . . . . . . . 1-5 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-6 2 Tutorial Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-3 Problems Covered by the Toolbox . . . . . . . . . . . . . . . . . . . . . . . . 2-3 Using the Optimization Functions . . . . . . . . . . . . . . . . . . . . . . . 2-5
Examples that Use Standard Algorithm 27 Unconstrained Minimization Example 2-8 Nonlinear Inequality Constrained Example 2-9 Constrained Example with Bounds 2-11 Constrained Example with gradients Gradient Check: Analytic Versus Numeric Equality Constrained Example 2-15 Maximization 2-16 Greater-Than-Zero Constraints 2-16 Additional Arguments: Avoiding Global Variables 2-16 Nonlinear Equations with analytic Jacobian Nonlinear Equations with Finite-Difference Jacobian 220 Multiobjective Examples Large-Scale Examples 2-33 Problems Covered by Large-Scale Methods Nonlinear Equations with Jacobian Nonlinear Equations with Jacobian Sparsity Pattern 239 Nonlinear Least-squares with Full Jacobian Sparsity Pattern Nonlinear minimization with gradient and hessian Nonlinear Minimization with gradient and Hessian Sparsity Pattern 2-44 Nonlinear Minimization with bound Constraints and banded Precondit 2-46 Nonlinear Minimization with Equality Constraint 2-50 Nonlinear minimization with a dense but structured Hessian and Equality Constraints 2-51 Quadratic Minimization with Bound Constraints Quadratic Minimization with a Dense but Structured Linear Least-Squares with Bound Constraints 2-60 Linear Programming with Equalities and Inequalities 2-61 Linear Programming with Dense Columns in the Equalities. 2-62 Default Parameter Settings 2-65 Changing the Default Settings 2-65 Displaying Iterative Output 2-68 Output Headings: Medium-Scale algorithms
ii Contents Examples that Use Standard Algorithms . . . . . . . . . . . . . . . . 2-7 Unconstrained Minimization Example . . . . . . . . . . . . . . . . . . . . 2-8 Nonlinear Inequality Constrained Example . . . . . . . . . . . . . . . 2-9 Constrained Example with Bounds . . . . . . . . . . . . . . . . . . . . . 2-11 Constrained Example with Gradients . . . . . . . . . . . . . . . . . . . 2-12 Gradient Check: Analytic Versus Numeric . . . . . . . . . . . . . . . 2-14 Equality Constrained Example . . . . . . . . . . . . . . . . . . . . . . . . . 2-15 Maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-16 Greater-Than-Zero Constraints . . . . . . . . . . . . . . . . . . . . . . . . 2-16 Additional Arguments: Avoiding Global Variables . . . . . . . . . 2-16 Nonlinear Equations with Analytic Jacobian . . . . . . . . . . . . . . 2-17 Nonlinear Equations with Finite-Difference Jacobian . . . . . . 2-20 Multiobjective Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-21 Large-Scale Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-33 Problems Covered by Large-Scale Methods . . . . . . . . . . . . . . . 2-34 Nonlinear Equations with Jacobian . . . . . . . . . . . . . . . . . . . . . 2-37 Nonlinear Equations with Jacobian Sparsity Pattern . . . . . . . 2-39 Nonlinear Least-Squares with Full Jacobian Sparsity Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-41 Nonlinear Minimization with Gradient and Hessian . . . . . . . 2-43 Nonlinear Minimization with Gradient and Hessian Sparsity Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-44 Nonlinear Minimization with Bound Constraints and Banded Preconditioner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-46 Nonlinear Minimization with Equality Constraints . . . . . . . . 2-50 Nonlinear Minimization with a Dense but Structured Hessian and Equality Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-51 Quadratic Minimization with Bound Constraints . . . . . . . . . . 2-55 Quadratic Minimization with a Dense but Structured Hessian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-57 Linear Least-Squares with Bound Constraints . . . . . . . . . . . . 2-60 Linear Programming with Equalities and Inequalities . . . . . . 2-61 Linear Programming with Dense Columns in the Equalities . 2-62 Default Parameter Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-65 Changing the Default Settings . . . . . . . . . . . . . . . . . . . . . . . . . 2-65 Displaying Iterative Output . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-68 Output Headings: Medium-Scale Algorithms . . . . . . . . . . . . . 2-68
Output Headings: Large-Scale algorithms Optimization of Inline Objects Instead of M-Files 2-74 Typical Problems and How to Deal with Them 276 Converting Your Code to Version 2 Syntax U t and optimget New Calling Sequences Example of Converting from constr to fmincon Selected Bibliography 2-92 Standard Algorithms 3 Optimization Overview Unconstrained Optimization Quasi-Newton Methods 36 Line search 38 Quasi-Newton Implementation Least-Squares Optimization 3-18 Gauss-Newton method 3-19 Levenberg- Marquardt Method Nonlinear Least-Squares Implementation Nonlinear Systems of Equations 3-24 Method 3-24 Trust-Region Dogleg Method 3-24 Nonlinear Equations Implementation Constrained Optimization .... 3-28 Sequential quadratic Programming(SQP) quadratic Programming(QP) Subproblem SQP Implementatie
iii Output Headings: Large-Scale Algorithms . . . . . . . . . . . . . . . 2-71 Optimization of Inline Objects Instead of M-Files . . . . . . . 2-74 Typical Problems and How to Deal with Them . . . . . . . . . . 2-76 Converting Your Code to Version 2 Syntax . . . . . . . . . . . . . 2-80 Using optimset and optimget . . . . . . . . . . . . . . . . . . . . . . . . . . 2-81 New Calling Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-81 Example of Converting from constr to fmincon . . . . . . . . . . . . 2-89 Selected Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-92 3 Standard Algorithms Optimization Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-3 Unconstrained Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-4 Quasi-Newton Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-6 Line Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-8 Quasi-Newton Implementation . . . . . . . . . . . . . . . . . . . . . . . . . 3-11 Least-Squares Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-18 Gauss-Newton Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-19 Levenberg-Marquardt Method . . . . . . . . . . . . . . . . . . . . . . . . . 3-20 Nonlinear Least-Squares Implementation . . . . . . . . . . . . . . . . 3-22 Nonlinear Systems of Equations . . . . . . . . . . . . . . . . . . . . . . . 3-24 Gauss-Newton Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-24 Trust-Region Dogleg Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-24 Nonlinear Equations Implementation . . . . . . . . . . . . . . . . . . . 3-26 Constrained Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-28 Sequential Quadratic Programming (SQP) . . . . . . . . . . . . . . . 3-29 Quadratic Programming (QP) Subproblem . . . . . . . . . . . . . . . 3-30 SQP Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-31
Multiobjective Optimization Introduction Goal Attainment method Algorithm Improvements for Goal Attainment Method Selected Bibliography 3-48 Large-Scale algorithms 4 Trust-Region Methods for Nonlinear Minimization 4-2 Preconditioned Conjugate Gradients 4-5 Linearly Constrained Problems Linear Equality Constraints Box Constraints Nonlinear Least-Squares 4-10 Quadratic Programming 4-11 Linear Least-Squares 4-12 Large-Scale Linear Programming Main algorithm 4-13 processing 4-16 Selected Bibliography 4-17
iv Contents Multiobjective Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 3-38 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-38 Goal Attainment Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-44 Algorithm Improvements for Goal Attainment Method . . . . . 3-45 Selected Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-48 4 Large-Scale Algorithms Trust-Region Methods for Nonlinear Minimization . . . . . . . 4-2 Preconditioned Conjugate Gradients . . . . . . . . . . . . . . . . . . . 4-5 Linearly Constrained Problems . . . . . . . . . . . . . . . . . . . . . . . . 4-7 Linear Equality Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-7 Box Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-7 Nonlinear Least-Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-10 Quadratic Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-11 Linear Least-Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-12 Large-Scale Linear Programming . . . . . . . . . . . . . . . . . . . . . 4-13 Main Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-13 Preprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-16 Selected Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-17
Function reference 5 Functions-By Category 52 Minimization E Least Squares(Curve Fitting) Utility 22333 Demos of Large-Scale Methods Demos of Medium-Scale methods Function Arguments 5-5 Optimization Options Parameters 5-10 Functions- Alphabetical List 5-15 Index
v 5 Function Reference Functions —By Category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-2 Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-2 Equation Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-2 Least Squares (Curve Fitting) . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-3 Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-3 Demos of Large-Scale Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 5-3 Demos of Medium-Scale Methods . . . . . . . . . . . . . . . . . . . . . . . . 5-4 Function Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-5 Optimization Options Parameters . . . . . . . . . . . . . . . . . . . . . 5-10 Functions — Alphabetical List . . . . . . . . . . . . . . . . . . . . . . . . 5-15 Index
Preface The preface consists of these sections Using This Guide(p viii) Explains the organization of this guide Related Products(p ix) Lists products that may be relevant to the kinds of tasks you can perform with the Optimization Toolbox. Typographical Conventions(p xi) Describes the typographical conventions used in this guide
Preface The Preface consists of these sections: Using This Guide (p. viii) Explains the organization of this guide. Related Products (p. ix) Lists products that may be relevant to the kinds of tasks you can perform with the Optimization Toolbox. Typographical Conventions (p. xi) Describes the typographical conventions used in this guide
Using This Guide This guide has the following chapters The"Introduction"introduces the Optimization Toolbox, explains technical conventions used in the book and lists features that are new in version 2.2 It also directs you to installation and configuration information ·The“ Tutorial” chapter shows you how to solve a variety of different ptimization problems. It includes a section that highlights large-scale problems. This chapter also provides information on how to use the toolbox functions in conjunction with Simulink using multiobjective optimization Other sections include information about changing default parameters and ·The“ Standard algorithms”and“ Large- Scale algorithms” chapters describe the algorithms used by the optimization functions Standard algorithms describes the problem formulations and algorithms for the medium-scale algorithms. "Large-Scale Algorithms" focuses on algorithms used to solve large sparse or structured problems The"Function Reference"chapter provides a detailed reference descriptio of each toolbox function. Reference descriptions include the functions syntax, a description of the different calling sequences available, and detailed information about arguments to the function, including relevant optimization options parameters Reference descriptions may also include examples, a summary of the function s algorithms, and references to additional reading material
Preface viii Using This Guide This guide has the following chapters: • The “Introduction” introduces the Optimization Toolbox, explains technical conventions used in the book, and lists features that are new in Version 2.2. It also directs you to installation and configuration information • The “Tutorial” chapter shows you how to solve a variety of different optimization problems. It includes a section that highlights large-scale problems. This chapter also provides information on how to use the toolbox functions in conjunction with Simulink using multiobjective optimization. Other sections include information about changing default parameters and using inline objects. • The “Standard Algorithms” and “Large-Scale Algorithms” chapters describe the algorithms used by the optimization functions. “Standard Algorithms” describes the problem formulations and algorithms for the medium-scale algorithms. “Large-Scale Algorithms” focuses on algorithms used to solve large sparse or structured problems. • The “Function Reference” chapter provides a detailed reference description of each toolbox function. Reference descriptions include the function’s syntax, a description of the different calling sequences available, and detailed information about arguments to the function, including relevant optimization options parameters. Reference descriptions may also include examples, a summary of the function’s algorithms, and references to additional reading material
Related products The Math Works provides several products that are relevant to the kinds of tasks you can perform with the Optimization Toolbox. For more information about any of these products, see either The online documentation for that product, if it is installed or if you are reading the documentation from the CD .TheMathWorksWebsiteatwww.mathworks.comseethe"productssection Note The toolboxes listed below all include functions that extend the capabilities of MATLAB. The blocksets all include blocks that extend the capabilities of Simulink. D Curve Fitting Toolbox Perform model fitting and analysis Data Acquisition Toolbox Acquire and send out data from plug-in data acquisition boards Database toolbox Exchange data with relational databases Financial Time series Analyze and manage financial time series data Financial Toolbox Model financial data and develop financial GARCH Toolbox Analyze financial volatility using univariate GARCH models LMI Control toolbox Design robust controllers using convex Neural Network Toolbox Design and simulate neural networks
Related Products ix Related Products The MathWorks provides several products that are relevant to the kinds of tasks you can perform with the Optimization Toolbox. For more information about any of these products, see either • The online documentation for that product, if it is installed or if you are reading the documentation from the CD • The MathWorks Web site at www.mathworks.com; see the “products” section Note The toolboxes listed below all include functions that extend the capabilities of MATLAB. The blocksets all include blocks that extend the capabilities of Simulink. Product Description Curve Fitting Toolbox Perform model fitting and analysis Data Acquisition Toolbox Acquire and send out data from plug-in data acquisition boards Database Toolbox Exchange data with relational databases Financial Time Series Toolbox Analyze and manage financial time series data Financial Toolbox Model financial data and develop financial analysis algorithms GARCH Toolbox Analyze financial volatility using univariate GARCH models LMI Control Toolbox Design robust controllers using convex optimization techniques Neural Network Toolbox Design and simulate neural networks
Product Description Nonlinear control Optimize design parameters in nonlinear Design Blockset control systems Signal Processing Perform signal processing, analysis, and Simulink Design and simulate continuous- and discrete-time systems Spline toolbox Create and manipulate spline approximation models of data Statistics toolbox Apply statistical algorithms and probability Symbolic/Extended Perform computations using symbolic Symbolic Math Toolbox mathematics and variable-precision arithmetic System Identification Create linear dynamic models from measured Toolbox input-output data
Preface x Nonlinear Control Design Blockset Optimize design parameters in nonlinear control systems Signal Processing Toolbox Perform signal processing, analysis, and algorithm development Simulink Design and simulate continuous- and discrete-time systems Spline Toolbox Create and manipulate spline approximation models of data Statistics Toolbox Apply statistical algorithms and probability models Symbolic/Extended Symbolic Math Toolbox Perform computations using symbolic mathematics and variable-precision arithmetic System Identification Toolbox Create linear dynamic models from measured input-output data Product Description
