## Problem-Based Workflow for Solving Equations

Note

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

To solve a system of equations, perform the following steps.

• Create an equation problem object by using `eqnproblem`. A problem object is a container in which you define equations. The equation problem object defines the problem and any bounds that exist in the problem variables.

For example, create an equation problem.

`prob = eqnproblem;`
• Create named variables by using `optimvar`. An optimization variable is a symbolic variable that you use to describe the equations. Include any bounds in the variable definitions.

For example, create a 15-by-3 array of variables named `'x'` with lower bounds of `0` and upper bounds of `1`.

`x = optimvar('x',15,3,'LowerBound',0,'UpperBound',1);`
• Define equations in the problem variables. For example:

```sumeq = sum(x,2) == 1; prob.Equations.sumeq = sumeq;```

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 for Optimization Variables and Expressions.

If necessary, include extra parameters in your equations as workspace variables; see Pass Extra Parameters in Problem-Based Approach.

• 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 equation-solving example with polynomials, see Solve Nonlinear System of Polynomials, Problem-Based. For a general nonlinear example, see Solve Nonlinear System of Equations, Problem-Based. For more extensive examples, see Systems of Nonlinear Equations.