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 a polynomial or rational expression, convert it to an optimization expression by using

`fcn2optimexpr`

. See Convert Nonlinear Function to Optimization Expression.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 a polynomial or rational expression, convert it to an optimization expression by using

`fcn2optimexpr`

. See Convert Nonlinear Function to Optimization Expression.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.

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.

`fcn2optimexpr`

| `optimoptions`

| `optimproblem`

| `optimvar`

| `show`

| `solve`

| `write`