# Optimization Toolbox

## Solve linear, quadratic, integer, and nonlinear optimization problems

Optimization Toolbox™ provides functions for finding parameters that minimize or maximize objectives while satisfying constraints. The toolbox includes solvers for linear programming (LP), mixed-integer linear programming (MILP), quadratic programming (QP), nonlinear programming (NLP), constrained linear least squares, nonlinear least squares, and nonlinear equations. You can define your optimization problem with functions and matrices or by specifying variable expressions that reflect the underlying mathematics.

You can use the toolbox solvers to find optimal solutions to continuous and discrete problems, perform tradeoff analyses, and incorporate optimization methods into algorithms and applications. The toolbox lets you perform design optimization tasks, including parameter estimation, component selection, and parameter tuning. It can be used to find optimal solutions in applications such as portfolio optimization, resource allocation, and production planning and scheduling.

### Get Started:

### Modeling

Transform a problem description into a mathematical form by defining optimization variables, an objective, and constraints, so that it can be solved with optimization techniques.

### Problem-Based Optimization

Write the objective and constraints in MATLAB using expressions that reflect the mathematical form. Build these with optimization variables, mathematical operators, matrix operations, and functions of the variables. Apply an automatically selected solver.

### Solver-Based Optimization

Write the objective and constraints in MATLAB using functions when nonlinear and coefficient matrices when linear. Build these with MATLAB matrix operations, mathematical operators, and functions. Apply an appropriate solver.

### Choosing a Solver

Solve using a nonlinear, quadratic, linear, mixed-integer, least-squares, or multiobjective solver depending on the types of objectives and constraints in the optimization problem. The solver is automatically selected in the problem-based approach.

### Setting Options

Set optimization options to tune the optimization process, for example, to choose the optimization algorithm used by the solver, or to set termination conditions. Set options to monitor and plot optimization solver progress.

### Reviewing and Improving Results

Review the optimality measures and iterative display to assess the solution. Use the links in the exit messages to learn more. Improve performance on nonlinear problems by supplying gradients or using parallel computing to estimate gradients.

### Solvers

Apply quasi-Newton, trust-region, or Nelder-Mead Simplex algorithms to solve unconstrained problems. Apply interior-point, sequential-quadratic-programming (SQP), or trust-region-reflective algorithms to solve constrained problems.

### Applications

Use nonlinear optimization for estimating and tuning parameters, finding optimal designs, computing optimal trajectories, constructing robust portfolios, and other applications where there is a nonlinear relationship between variables.

### Linear Programming Solvers

Apply dual-simplex or interior-point algorithms to solve linear programs.

### Quadratic Programming Solvers

Apply interior-point, active-set, or trust-region-reflective algorithms to solve quadratic programs.

### Applications

Use linear programming on problems such as resource allocation, production planning, blending, and investment planning. Use quadratic programming on problems such as design optimization, portfolio optimization, and control of hydroelectric dams.

### Solvers

Solve mixed-integer linear programming problems using the branch and bound algorithm, which includes preprocessing, heuristics for generating feasible points, and cutting planes.

### Mixed-Integer Linear Programming-Based Algorithms

Use the mixed-integer linear programming solver to build special-purpose algorithms.

### Applications

Model with integer variables when there are on/off decisions or logical constraints as well as when variable values must be integral. Routing, scheduling, planning, assignment, and capital budgeting problems are typical applications.

### Solvers

Formulate problems as either goal-attainment or minimax. Use goal-attainment when there are optionally weighted goal values for each of the objectives. Use minimax to minimize the worst-case value of a set of objective functions.

### Applications

Use multiobjective optimization when tradeoffs are required for two or more conflicting objectives. Examples are weight and strength in structural design and risk and return in portfolio optimization.

### Solvers

Apply Levenberg-Marquardt, trust-region, active-set, or interior-point algorithms.

### Linear Least-Squares Applications

Use linear least-squares solvers to fit a linear model to acquired data or to solve a system of linear equations, including when the parameters are subject to bound and linear constraints.

### Nonlinear Least-Squares Applications

Use nonlinear least-squares solvers to fit a nonlinear model to acquired data or to solve a system of nonlinear equations, including when the parameters are subject to bound constraints.

### MATLAB Compiler Support

Use MATLAB Compiler™ and MATLAB Compiler SDK™ to deploy MATLAB optimization models as standalone executables, web apps, C/C++ shared libraries, Microsoft^{®} .NET assemblies, Java^{®} classes, and Python^{®} packages.

### Code Generation

Generate portable and readable C or C++ code to solve optimization problems using MATLAB Coder^{TM}. Compile the generated code for any hardware, including embedded systems.

### Code Generation

Generate C/C++ code to solve quadratic programming problems with `quadprog`

(requires MATLAB Coder)

### Code Generation

Generate C/C++ code to solve nonlinear constrained optimization problems with `fmincon`

(requires MATLAB Coder)

### Optimization Modeling

Define nonlinear optimization objective functions or constraints using elementary functions of optimization variables

### Quadratic Programming and Linear Least Squares

Solve dense problems quickly and accurately using the active-set algorithm

### Mixed-Integer Linear Programming

Obtain feasible points faster with additional heuristics

### Infeasibility Analysis Example

Identify conflicting linear constraints by finding irreducible infeasible or maximal feasible subsets

See release notes for details on any of these features and corresponding functions.