## Smooth Formulations of Nonsmooth Functions

To smooth an otherwise nonsmooth problem, you can sometimes add auxiliary variables. For example,

f(x) = max(g(x),h(x))

can be a nonsmooth function even when g(x) and h(x) are smooth, as illustrated by the following functions.

$\begin{array}{l}g\left(x\right)=\mathrm{sin}\left(x\right)\\ h\left(x\right)=\mathrm{cos}\left(x\right)\\ f\left(x\right)=\mathrm{max}\left(g\left(x\right),h\left(x\right)\right).\end{array}$

f(x) is nonsmooth at the points x = π/4 and x = 5π/4.

This lack of smoothness can cause problems for Optimization Toolbox™ solvers, all of which assume that objective functions and nonlinear constraint functions are continuously differentiable. So, if you try to solve

x = mint(f(t)) starting from the point x0 = 1,

you do not get an exit flag of 1, because the solution is not differentiable at the locally minimizing point x = π/4.

fun1 = @sin;
fun2 = @cos;
fun = @(x)max(fun1(x),fun2(x));
[x1,fval1,eflag1] = fminunc(fun,1)
Local minimum possible.

fminunc stopped because it cannot decrease the objective function
along the current search direction.

<stopping criteria details>

x1 =

0.7854

fval1 =

0.7071

eflag1 =

5

Sometimes, you can use an auxiliary variable to turn a nonsmooth problem into a smooth problem. For the previous example, consider the auxiliary variable y with the smooth constraints

$\begin{array}{l}y\ge g\left(x\right)\\ y\ge h\left(x\right).\end{array}$

Consider the optimization problem, subject to these constraints,

$\underset{x}{\mathrm{min}}y.$

The resulting solution x, y is the solution to the original problem

$\underset{x}{\mathrm{min}}f\left(x\right)=\underset{x}{\mathrm{min}}\mathrm{max}\left(g\left(x\right),h\left(x\right)\right).$

This formulation uses the problem-based approach.

myvar = optimvar("myvar");
auxvar = optimvar("auxvar");
smprob = optimproblem("Objective",auxvar);
smprob.Constraints.cons1 = auxvar >= sin(myvar);
smprob.Constraints.cons2 = auxvar >= cos(myvar);
x0.myvar = 1;
x0.auxvar = 1;
[sol2,fval2,eflag2] = solve(smprob,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.

sol2 =

struct with fields:

auxvar: 0.7071
myvar: 0.7854

fval2 =

0.7071

eflag2 =

OptimalSolution

This same concept underlies the formulation of the fminimax function; see Goal Attainment Method.