Overview

In mathematics, computer science and operations research, mathematical optimization, also spelled mathematical optimisation (alternatively named mathematical programming or simply optimization or optimisation), is the selection of a best element (with regard to some criterion) from some set of available alternatives. In the simplest case, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. The generalization of optimization theory and techniques to other formulations comprises a large area of applied mathematics. More generally, optimization includes finding best available values of some objective function given a defined domain (or input), including a variety of different types of objective functions and different types of domains. MATLAB Optimization Toolbox provides functions for finding parameters that minimize or maximize objectives while satisfying constraints. The toolbox includes solvers for linear programming, mixed-integer linear programming, quadratic programming, nonlinear optimization, and nonlinear least squares. You can use these solvers to find optimal solutions to continuous and discrete problems, perform tradeoff analyses, and incorporate optimization methods into algorithms and applications. This book develops the following topics: - Linear Programming - Nonlinear Programming - Constrained Linear and Nonlinear Problem - Optimization Toolbox Solvers - Optimization Decision Table - fmincon Algorithms - fsolve Algorithms - fminunc Algorithms - Least Squares Algorithms - Linear Programming Algorithms - Quadratic Programming Algorithms - Large-Scale vs. Medium-Scale Algorithms - Potential Inaccuracy with Interior-Point Algorithms - Edit Optimization Parameters - Complex Numbers in Optimization Toolbox Solvers - Scalar Objective Functions - Vector and Matrix Objective Functions - Objective Functions for Linear or Quadratic Problems - Maximizing an Objective - Bound Constraints - Linear an Nonlinlear Constraints - optimoptions and optimset - Tolerances and Stopping Criteria - Checking Validity of Gradients or Jacobians - Iterations and Function Counts - First-Order Optimality Measure - Lagrange Multiplier Structures - Plot an Optimization During Execution - Local vs. Global Optima - Optimizing a Simulation or Ordinary Differential Equation - Optimization App - Nonlinear algorithms and examples - Unconstrained Nonlinear Optimization Algorithms - fminsearch Algorithm - fminunc Unconstrained Minimization - Minimization with Gradient and Hessian - Minimization with Gradient and Hessian Sparsity Pattern - Constrained Nonlinear Optimization Algorithms - Nonlinear Inequality Constraints - Nonlinear Constraints with Gradients - fmincon Interior-Point Algorithm with Analytic Hessian - Linear or Quadratic Objective with Quadratic Constraints - Nonlinear Equality and Inequality Constraints - Optimization App with the fmincon Solver - Minimization with Bound Constraints and Banded Preconditioner - Minimization with Linear Equality Constraints - Minimization with Dense Structured Hessian, Linear Equalities - One-Dimensional Semi-Infinite Constraints - Two-Dimensional Semi-Infinite Constraint

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