book

Solving Optimization Problems with MATLAB®

Name: Solving Optimization Problems with MATLAB®
ISBN: 9783110663693

by Dingyü Xue, Tsinghua University Tsinghua University Press

April 2020

Intermediate to advanced

342 pages

10h 53m

English

De Gruyter

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Preface
1 An introduction to equations and optimization problems
1.1 Equations and their solutions1.2 Origins and development of optimization problems1.3 Structure of the book1.4 Exercises
2 Solutions of algebraic equations
2.1 Solutions of polynomial equations2.1.1 Polynomial equations of degrees 1 and 22.1.2 Analytical solutions of cubic equations2.1.3 Analytical solutions of quartic equation2.1.4 Higher-degree equations and Abel–Ruffini theorem2.2 Graphical methods for nonlinear equations2.2.1 Smooth graphics for implicit functions2.2.2 Univariate equations2.2.3 Equations with two unknowns2.2.4 Isolated equation solutions2.3 Numerical solutions of algebraic equations2.3.1 Newton–Raphson iterative algorithm2.3.2 Direct solution methods with MATLAB2.3.3 Accuracy specifications2.3.4 Complex domain solutions2.4 Accurate solutions of simultaneous equations2.4.1 Analytical solutions of low-degree polynomial equations2.4.2 Quasianalytical solutions of polynomial-type equations2.4.3 Quasianalytical solutions of polynomial matrix equations2.4.4 Quasianalytical solutions of nonlinear equations2.5 Nonlinear matrix equations with multiple solutions2.5.1 An equation solution idea and its implementation2.5.2 Pseudopolynomial equations2.5.3 A quasianalytical solver2.6 Underdetermined algebraic equations2.7 Exercises
3 Unconstrained optimization problems
3.1 Introduction to unconstrained optimization problems3.1.1 The mathematical model of unconstrained optimization problems3.1.2 Analytical solutions of unconstrained minimization problems3.1.3 Graphical solutions3.1.4 Local and global optimum solutions3.1.5 MATLAB implementation of optimization algorithms3.2 Direct solutions of unconstrained optimization problem with MATLAB3.2.1 Direct solution methods3.2.2 Control options in optimization3.2.3 Additional parameters3.2.4 Intermediate solution process3.2.5 Structured variable description of optimization problems3.2.6 Gradient information3.2.7 Optimization solutions from scattered data3.2.8 Parallel computation in optimization problems3.3 Towards global optimum solutions3.4 Optimization with decision variable bounds3.4.1 Univariate optimization problem3.4.2 Multivariate optimization problems3.4.3 Global optimum solutions3.5 Application examples of optimization problems3.5.1 Solutions of linear regression problems3.5.2 Least-squares curve fitting3.5.3 Shooting method in boundary value differential equations3.5.4 Converting algebraic equations into optimization problems3.6 Exercises
4 Linear and quadratic programming
4.1 An introduction to linear programming4.1.1 Mathematical model of linear programming problems4.1.2 Graphical solutions of linear programming problems4.1.3 Introduction to the simplex method4.2 Direct solutions of linear programming problems4.2.1 A linear programming problem solver4.2.2 Linear programming problems with multiple decision vectors4.2.3 Linear programming with double subscripts4.2.4 Transportation problem4.3 Problem-based description and solution of linear programming problems4.3.1 MPS file for linear programming problems4.3.2 Problem-based description of linear programming problems4.3.3 Conversions in linear programming problems4.4 Quadratic programming4.4.1 Mathematical quadratic programming models4.4.2 Direct solutions of quadratic programming problems4.4.3 Problem-based quadratic programming problem description4.4.4 Quadratic programming problem with double subscripts4.5 Linear matrix inequalities4.5.1 Description of linear matrix inequality problems4.5.2 Lyapunov inequalities4.5.3 Classifications of LMI problems4.5.4 MATLAB solutions of LMI problems4.5.5 Optimization solutions with YALMIP toolbox4.5.6 Trials on nonconvex problems4.5.7 Problems with quadratic constraints4.6 Exercises
5 Nonlinear programming
5.1 Introduction to nonlinear programming5.1.1 Mathematical models of nonlinear programming problems5.1.2 Feasible regions and graphical methods5.1.3 Examples of numerical methods5.2 Direct solutions of nonlinear programming problems5.2.1 Direct solution using MATLAB5.2.2 Handling of earlier termination phenomenon5.2.3 Gradient information5.2.4 Solving problems with multiple decision vectors5.2.5 Complicated nonlinear programming problems5.3 Trials with global nonlinear programming solver5.3.1 Trials on global optimum solutions5.3.2 Nonconvex quadratic programming problems5.3.3 Concave-cost transportation problem5.3.4 Testing of the global optimum problem solver5.3.5 Handling piecewise objective functions5.4 Bilevel programming problems5.4.1 Bilevel linear programming problems5.4.2 Bilevel quadratic programming problem5.4.3 Bilevel program solutions with YALMIP Toolbox5.5 Nonlinear programming applications5.5.1 Maximum inner polygon inside a circle5.5.2 Semiinfinite programming problems5.5.3 Pooling and blending problem5.5.4 Optimization design of heat exchange network5.5.5 Solving nonlinear equations with optimization techniques5.6 Exercises
6 Mixed integer programming
6.1 Introduction to integer programming6.1.1 Integer and mixed-integer programming problems6.1.2 Computational complexity of integer programming problems6.2 Enumeration methods for integer programming6.2.1 An introduction to the enumeration method6.2.2 Discrete programming6.2.3 0−1 programming6.2.4 Trials on mixed-integer programming problems6.3 Solutions of mixed-integer programming problems6.3.1 Mixed-integer linear programming6.3.2 Integer programming with YALMIP Toolbox6.3.3 Mixed-integer nonlinear programming6.3.4 A class of discrete programming problems6.3.5 Solutions of ordinary discrete programming problems6.4 Mixed 0−1 programming problems6.4.1 0–1 linear programming problems6.4.2 0–1 nonlinear programming problems6.5 Mixed-integer programming applications6.5.1 Optimal material usage6.5.2 Assignment problem6.5.3 Traveling salesman problem6.5.4 Knapsack problems6.5.5 Sudoku problems6.6 Exercises
7 Multiobjective programming
7.1 Introduction to multiobjective programming7.1.1 Background introduction7.1.2 Mathematical model of multiobjective programming7.1.3 Graphical solution of multiobjective programming problems7.2 Multiobjective programming conversions and solutions7.2.1 Least-squares solutions of multiobjective programming problems7.2.2 Linear weighting conversions7.2.3 Best compromise solution of linear programs7.2.4 Least-squares linear programming7.3 Pareto optimal solutions7.3.1 Nonuniqueness of multiobjective programming7.3.2 Dominant solutions and Pareto frontiers7.3.3 Computations of Pareto frontier7.4 Minimax problems7.5 Exercises
8 Dynamic programming and shortest paths
8.1 An introduction to dynamic programming8.1.1 Concept and mathematical models in dynamic programming8.1.2 Dynamic programming solutions of linear programming problems8.2 Shortest path problems in oriented graphs8.2.1 Examples of oriented graphs8.2.2 Manual solutions of shortest path problem8.2.3 Solution with dynamic programming formulation8.2.4 Matrix representation of graphs8.2.5 Finding the shortest path8.2.6 Dijkstra algorithm implementation8.3 Optimal paths for undigraphs8.3.1 Matrix description8.3.2 Route planning for cities with absolute coordinates8.4 Exercises
9 Introduction to intelligent optimization methods
9.1 Intelligent optimization algorithms9.1.1 Genetic algorithms9.1.2 Particle swarm optimization methods9.2 MATLAB Global Optimization Toolbox9.3 Examples and comparative studies of intelligent optimization methods9.3.1 Unconstrained optimization problems9.3.2 Constrained optimization problems9.3.3 Mixed-integer programming9.3.4 Discrete programming problems with the genetic algorithm9.4 Exercises

Bibliography
Subject Index
Sachregister

Overview

This book focuses on solving optimization problems with MATLAB. Descriptions and solutions of nonlinear equations of any form are studied first. Focuses are made on the solutions of various types of optimization problems, including unconstrained and constrained optimizations, mixed integer, multiobjective and dynamic programming problems. Comparative studies and conclusions on intelligent global solvers are also provided.

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