## Problem-Based Optimization Workflow

### Note

Optimization Toolbox™ provides two approaches for solving single-objective optimization problems. This topic describes the problem-based approach. Solver-Based Optimization Problem Setup describes the solver-based approach.

To solve an optimization problem, perform the following steps.

• Create an optimization problem object by using `optimproblem`. A problem object is a container in which you define an objective expression and constraints. The optimization problem object defines the problem and any bounds that exist in the problem variables.

For example, create a maximization problem.

`prob = optimproblem('ObjectiveSense','maximize');`
• Create named variables by using `optimvar`. An optimization variable is a symbolic variable that you use to describe the problem objective and constraints. Include any bounds in the variable definitions.

For example, create a 15-by-3 array of binary variables named `'x'`.

`x = optimvar('x',15,3,'Type','integer','LowerBound',0,'UpperBound',1);`
• Define the objective function in the problem object as an expression in the named variables.

### Note

If you have a nonlinear function that is not composed of polynomials, rational expressions, and elementary functions such as `exp`, then convert the function to an optimization expression by using `fcn2optimexpr`. See Convert Nonlinear Function to Optimization Expression and Supported Operations on Optimization Variables and Expressions.

For example, assume that you have a real matrix `f` of the same size as a matrix of variables `x`, and the objective is the sum of the entries in `f` times the corresponding variables `x`.

`prob.Objective = sum(sum(f.*x));`
• Define constraints for optimization problems as either comparisons in the named variables or as comparisons of expressions.

### Note

If you have a nonlinear function that is not composed of polynomials, rational expressions, and elementary functions such as `exp`, then convert the function to an optimization expression by using `fcn2optimexpr`. See Convert Nonlinear Function to Optimization Expression and Supported Operations on Optimization Variables and Expressions.

For example, assume that the sum of the variables in each row of `x` must be one, and the sum of the variables in each column must be no more than one.

```onesum = sum(x,2) == 1; vertsum = sum(x,1) <= 1; prob.Constraints.onesum = onesum; prob.Constraints.vertsum = vertsum;```
• For nonlinear problems, set an initial point as a structure whose fields are the optimization variable names. For example:

```x0.x = randn(size(x)); x0.y = eye(4); % Assumes y is a 4-by-4 variable```
• Solve the problem by using `solve`.

```sol = solve(prob); % Or, for nonlinear problems, sol = solve(prob,x0)```

In addition to these basic steps, you can review the problem definition before solving the problem by using `show` or `write`. Set options for `solve` by using `optimoptions`, as explained in Change Default Solver or Options.

### Warning

The problem-based approach does not support complex values in an objective function, nonlinear equalities, or nonlinear inequalities. If a function calculation has a complex value, even as an intermediate value, the final result can be incorrect.

For a basic mixed-integer linear programming example, see Mixed-Integer Linear Programming Basics: Problem-Based or the video version Solve a Mixed-Integer Linear Programming Problem Using Optimization Modeling. For a nonlinear example, see Solve a Constrained Nonlinear Problem, Problem-Based. For more extensive examples, see Problem-Based Nonlinear Optimization, Linear Programming and Mixed-Integer Linear Programming, or Quadratic Programming.