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Solver-Based Nonlinear Optimization

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

See First Choose Problem-Based or Solver-Based Approach for choosing between problem-based optimization and solver-based optimization.

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


fminbndFind minimum of single-variable function on fixed interval
fminconFind minimum of constrained nonlinear multivariable function
fminsearchFind minimum of unconstrained multivariable function using derivative-free method
fminuncFind minimum of unconstrained multivariable function
fseminfFind minimum of semi-infinitely constrained multivariable nonlinear function


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.

fminunc Unconstrained Minimization

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 the Optimization Toolbox™

Tutorial example showing how to solve nonlinear problems and pass extra parameters.

Optimization App with the fmincon Solver

Example of nonlinear programming with constraints using the Optimization app.

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.

Minimization with Bound Constraints and Banded Preconditioner

Example showing efficiency gains possible with structured nonlinear problems.

Minimization with Linear Equality Constraints

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.

Symbolic Math Toolbox Calculates Gradients and Hessians

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.

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?

Using multiple processors for optimization.

Using Parallel Computing in Optimization Toolbox

Automatic gradient estimation in parallel.

Improving Performance with Parallel Computing

Considerations 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

Describes optimization options.

Local vs. Global Optima

Explains why solvers might not find the smallest minimum.


Lists published materials that support concepts implemented in the solver algorithms.