Solve Nonlinear Feasibility Problem, Problem-Based

This example shows how to find a point that satisfies all the constraints in a problem, with no objective function to minimize.

Problem Definition

For example, suppose that you have the following constraints:

`$\begin{array}{l}\left(y+{x}^{2}{\right)}^{2}+0.1{y}^{2}\le 1\\ y\le \mathrm{exp}\left(-x\right)-3\\ y\le x-4.\end{array}$`

Do any points $\left(x,y\right)$ satisfy all of the constraints?

Problem-Based Solution

Create an optimization problem that has only constraints, no objective function.

```x = optimvar('x'); y = optimvar('y'); prob = optimproblem; cons1 = (y + x^2)^2 + 0.1*y^2 <= 1; cons2 = y <= exp(-x) - 3; cons3 = y <= x - 4; prob.Constraints.cons1 = cons1; prob.Constraints.cons2 = cons2; prob.Constraints.cons3 = cons3; show(prob)```
``` OptimizationProblem : Solve for: x, y minimize : subject to cons1: ((y + x.^2).^2 + (0.1 .* y.^2)) <= 1 subject to cons2: y <= (exp(-x) - 3) subject to cons3: y - x <= -4 ```

Create a pseudorandom start point structure `x0` with fields `x` and `y` for the optimization variables.

```rng default x0.x = randn; x0.y = randn;```

Solve the problem starting from `x0`.

`[sol,~,exitflag,output] = solve(prob,x0)`
```Solving problem using fmincon. Local minimum found that satisfies the constraints. Optimization completed because the objective function is non-decreasing in feasible directions, to within the value of the optimality tolerance, and constraints are satisfied to within the value of the constraint tolerance. ```
```sol = struct with fields: x: 1.7903 y: -3.0102 ```
```exitflag = OptimalSolution ```
```output = struct with fields: iterations: 6 funcCount: 9 constrviolation: 0 stepsize: 0.2906 algorithm: 'interior-point' firstorderopt: 0 cgiterations: 0 message: '...' bestfeasible: [1x1 struct] objectivederivative: "closed-form" constraintderivative: "forward-AD" solver: 'fmincon' ```

The solver finds a feasible point.

Initial Point Importance

The solver can fail to find a solution when starting from some initial points. Set the initial point `x0.x = -1`, `x0.y = -4` and solve the problem starting from `x0`.

```x0.x = -1; x0.y = -4; [sol2,~,exitflag2,output2] = solve(prob,x0)```
```Solving problem using fmincon. Converged to an infeasible point. fmincon stopped because it is unable to find a point locally that satisfies the constraints within the value of the constraint tolerance. ```
```sol2 = struct with fields: x: -2.1266 y: -4.6657 ```
```exitflag2 = NoFeasiblePointFound ```
```output2 = struct with fields: iterations: 121 funcCount: 279 constrviolation: 1.4609 stepsize: 1.9997e-10 algorithm: 'interior-point' firstorderopt: 0 cgiterations: 261 message: '...' bestfeasible: [] objectivederivative: "closed-form" constraintderivative: "forward-AD" solver: 'fmincon' ```

Check the infeasibilities at the returned point.

`inf1 = infeasibility(cons1,sol2)`
```inf1 = 1.1974 ```
`inf2 = infeasibility(cons2,sol2)`
```inf2 = 0 ```
`inf3 = infeasibility(cons3,sol2)`
```inf3 = 1.4609 ```

Both `cons1` and `cons3` are infeasible at the solution `sol2`. The results highlight the importance of using multiple start points to investigate and solve a feasibility problem.

Visualize Constraints

To visualize the constraints, plot the points where each constraint function is zero by using `fimplicit`. The `fimplicit` function passes numeric values to its functions, whereas the `evaluate` function requires a structure. To tie these functions together, use the `evaluateExpr` helper function, which appears at the end of this example. This function simply puts passed values into a structure with the appropriate names.

Note: If you use the live script file for this example, the `evaluateExpr` function is already included at the end of the file. Otherwise, you need to create this function at the end of your .m file or add it as a file on the MATLAB® path.

Avoid a warning that occurs because the `evaluateExpr` function does not work on vectorized inputs.

```s = warning('off','MATLAB:fplot:NotVectorized'); cc1 = (y + x^2)^2 + 0.1*y^2 - 1; fimplicit(@(a,b)evaluateExpr(cc1,a,b),[-2 2 -4 2],'r') hold on cc2 = y - exp(-x) + 3; fimplicit(@(a,b)evaluateExpr(cc2,a,b),[-2 2 -4 2],'k') cc3 = y - x + 4; fimplicit(@(x,y)evaluateExpr(cc3,x,y),[-2 2 -4 2],'b') hold off```

`warning(s);`

The feasible region is inside the red outline and below the black and blue lines. The feasible region is at the lower right of the red outline.

Helper Function

This code creates the `evaluateExpr` helper function.

```function p = evaluateExpr(expr,x,y) pt.x = x; pt.y = y; p = evaluate(expr,pt); end```