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.

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

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

 Code for Creating the Figure

x = linspace(0,2*pi);
g = @sin;
h = @cos;
f = @(x)max(g(x),h(x));
plot(x,g(x),'b.-',x,h(x),'r.-')
hold on
plot(x,f(x),'k--',"LineWidth",2)
t = [pi,5*pi]/4;
plot(t,sin(t),'mo',"MarkerSize",15)
legend("sin","cos","max","nonsmooth","Location","north")
hold off

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 = min_t(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

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

Consider the optimization problem, subject to these constraints,

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

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

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

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.