Solver-Based Nonlinear Optimization

Solve nonlinear minimization and semi-infinite programming problems in serial or parallel using the solver-based approach

Before you begin to solve an optimization problem, you must choose the appropriate approach: problem-based or solver-based. For details, see First Choose Problem-Based or Solver-Based Approach.

For problem setup, see Solver-Based Optimization Problem Setup.

Functions

`fminbnd`	Find minimum of single-variable function on fixed interval
`fmincon`	Find minimum of constrained nonlinear multivariable function
`fminsearch`	Find minimum of unconstrained multivariable function using derivative-free method
`fminunc`	Find minimum of unconstrained multivariable function
`fseminf`	Find minimum of semi-infinitely constrained multivariable nonlinear function
`checkGradients`	Check first derivative function against finite-difference approximation (Since R2023b)
`optim.coder.infbound`	Infinite bound support for code generation (Since R2022b)

Live Editor Tasks

Optimize

Optimize or solve equations in the Live Editor (Since R2020b)

Topics

Unconstrained Solver-Based Applications

Banana Function Minimization
Shows how to solve for the minimum of Rosenbrock's function using different solvers, with or without gradients.
Solve Nonlinear Problem with Many Variables
Choose appropriate options for large nonlinear problems.
Unconstrained Minimization Using fminunc
Example of unconstrained nonlinear programming.
Minimization with Gradient and Hessian
Example of unconstrained nonlinear programming including derivatives.
Minimization with Gradient and Hessian Sparsity Pattern
Example of nonlinear programming using some derivative information.

Constrained Solver-Based Applications

Tutorial for Optimization Toolbox
Tutorial example showing how to solve nonlinear problems and pass extra parameters.
Optimize Live Editor Task with fmincon Solver
Example of nonlinear programming with constraints using the Optimize Live Editor Task.
Nonlinear Inequality Constraints
Example of nonlinear programming with nonlinear inequality constraints.
Nonlinear Constraints with Gradients
Example of nonlinear programming with derivative information.
fmincon Interior-Point Algorithm with Analytic Hessian
Example of nonlinear programming with all derivative information.
Linear or Quadratic Objective with Quadratic Constraints
This example shows how to solve an optimization problem that has a linear or quadratic objective and quadratic inequality constraints.
Nonlinear Equality and Inequality Constraints
Nonlinear programming with both types of nonlinear constraints.
How to Use All Types of Constraints
Example showing all constraints.
Obtain Best Feasible Point
Find the best feasible point in the output structure.
Solve Nonlinear Problem with Many Variables
Choose appropriate options for large nonlinear problems.
Minimization with Bound Constraints and Banded Preconditioner
Example showing efficiency gains possible with structured nonlinear problems.
Minimization with Linear Equality Constraints, Trust-Region Reflective Algorithm
Example showing nonlinear programming with only linear equality constraints.
Minimization with Dense Structured Hessian, Linear Equalities
Example showing how to save memory in nonlinear programming with a structured Hessian and only linear equality constraints or only bounds.
Calculate Gradients and Hessians Using Symbolic Math Toolbox
Example showing how to calculate derivatives symbolically for optimization solvers.
Using Symbolic Mathematics with Optimization Toolbox Solvers
Use Symbolic Math Toolbox™ to generate gradients and Hessians.

Code Generation

Code Generation in fmincon Background
Prerequisites to generate C code for nonlinear optimization.
Code Generation for Optimization Basics
Learn the basics of code generation for the fmincon optimization solver.
Static Memory Allocation for fmincon Code Generation
Use static memory allocation in code generation when the problem changes.
Optimization Code Generation for Real-Time Applications
Explore techniques for handling real-time requirements in generated code.

Semi-Infinite Constraints

One-Dimensional Semi-Infinite Constraints
Example showing how to use one-dimensional semi-infinite constraints in nonlinear programming.
Two-Dimensional Semi-Infinite Constraint
Example showing how to use two-dimensional semi-infinite constraints in nonlinear programming.
Analyzing the Effect of Uncertainty Using Semi-Infinite Programming
This example shows how to use semi-infinite programming to investigate the effect of uncertainty in the model parameters of an optimization problem.

Parallel Computing

What Is Parallel Computing in Optimization Toolbox?
Use multiple processors for optimization.
Using Parallel Computing in Optimization Toolbox
Perform gradient estimation in parallel.
Improving Performance with Parallel Computing
Investigate factors for speeding optimizations.
Minimizing an Expensive Optimization Problem Using Parallel Computing Toolbox
Example showing how to use parallel computing in both Global Optimization Toolbox and Optimization Toolbox™ solvers.

Simulation or ODE

Optimizing a Simulation or Ordinary Differential Equation
Special considerations in optimizing simulations, black-box objective functions, or ODEs.

Algorithms and Other Theory

Unconstrained Nonlinear Optimization Algorithms
Minimizing a single objective function in n dimensions without constraints.
Constrained Nonlinear Optimization Algorithms
Minimizing a single objective function in n dimensions with various types of constraints.
fminsearch Algorithm
Steps that fminsearch takes to minimize a function.
Optimization Options Reference
Explore optimization options.
Local vs. Global Optima
Explains why solvers might not find the smallest minimum.
Smooth Formulations of Nonsmooth Functions
Reformulate some nonsmooth functions as smooth functions by using auxiliary variables.
Optimization Bibliography
Lists published materials that support concepts implemented in the optimization solver algorithms.